[Question]
  [
## Introductions
A 2×n Boolean matrix can be represented as a string of the four characters `. ':`.
The string has an "upper row" and a "lower row", with dots representing 1s and empty spaces representing 0s.
For example, the 2×6 matrix
```
1 0 1 0 0 1
0 0 0 1 0 1
```
can be represented as `' '. :`.
Your task is to take a matrix in this "compressed form", and rotate its entries one step clockwise, like a conveyor belt.
## Input
Your input is a single string over the characters `. ':`.
Its length is guaranteed to be at least 2.
## Output
Your output shall be the input string, but with every dot rotated one step in the clockwise direction.
More explicitly, the dots on the upper row more one place to the right, except the rightmost one, which moves down.
The dots on the lower row move one step to the left, except the leftmost one, which moves up.
In particular, the output string must have the same length as the original, and whitespace is significant.
## Example
Consider the input string `:..:'.`, which corresponds to the 2×6 matrix
```
1 0 0 1 1 0
1 1 1 1 0 1
```
The rotated version of this matrix is
```
1 1 0 0 1 1
1 1 1 0 1 0
```
which corresponds to the string `::. :'`.
## Rules and scoring
You can write a full program or a function.
The lowest byte count wins, and standard loopholes are disallowed.
You can decide whether the input and output are enclosed in quotes, and one trailing newline is also acceptable in both.
## Test cases
These test cases are enclosed in double quotes.
```
"  " -> "  "
" ." -> ". "
". " -> "' "
"' " -> " '"
" '" -> " ."
": " -> "''"
"''" -> " :"
":." -> ":'"
":.'" -> ":'."
"..." -> ":. "
": :" -> "':."
"':." -> ".:'"
".:'" -> ": :"
"    " -> "    "
"::::" -> "::::"
":..:'." -> "::. :'"
" :  .:'" -> ". '.. :"
": ''. :" -> "'' :'.."
"........" -> ":...... "
"::::    " -> ":::''   "
"    ::::" -> "   ..:::"
" : : : : " -> ". : : : '"
".'.'.'.'.'" -> "'.'.'.'.'."
".. ::  '.' :." -> ": ..'' .' :.'"
".'  '.::  :.'. . ::.'  '. . .::'  :.'." -> "' ' .:.''..'.'. ..:' ' .'. ...'''..'.'"
```
  
[Answer]
# JavaScript (ES6), ~~100~~ ~~97~~ 93 bytes
*Saved 4 bytes thanks to @edc65*
```
s=>s.replace(/./g,(c,i)=>" '.:"[(i?q>' '&q!=".":c>"'")+(q=c,s[++i]?s[i]>"'":c>' '&c!=".")*2])
```
### How it works
This decides on the character we need to insert by performing some calculations on the chars before and after the current one. We sum:
* If it's the first char, and it has a dot on the bottom, 2;
* Otherwise, if the one before it has a dot on top, 2.
* If it's the last char, and it has a dot on top, 1;
* Otherwise, if the one after it has a dot on the bottom, 1.
This sums nicely to 0 for a space, 1 for `'`, 2 for `.`, and 3 for `:`.
## Test snippet
```
f=s=>s.replace(/./g,(c,i)=>" '.:"[(i?q>' '&q!=".":c>"'")+(q=c,s[++i]?s[i]>"'":c>' '&c!=".")*2])
```
```
Try it out:
<input id=A type="text" value=".'  '.::  :.'. . ::.'  '. . .::'  :.'.">
<button onclick="B.innerHTML=f(A.value)">Run</button>
<pre id=B></pre>
```
[Answer]
# Perl, ~~70~~ ~~69~~ ~~64~~ ~~63~~ ~~61~~ 60 bytes
Includes +2 for `-lp`
Run with the input string on STDIN, e.g.
```
perl -lp rotatedots.pl <<< ":..:'."
```
`rotatedots.pl`:
```
y/'.:/02/r=~/./;y/.':/01/;$_=$'.2*chop|$&/2 .$_;y;0-3; '.:
```
## Explanation
```
y/'.:/02/r                                        Construct bottom row but
                                                  with 2's instead of 1's
                                                  Return constructed value
                                                  (for now assume space
                                                  becomes 0 too)
          =~/./                                   Match first digit on bottom
                                                  row into $&. $' contains
                                                  the rest of the bottom row
                y/.':/01/                         Convert $_ to top row
                                                  (again assume space
                                                  becomes 0 too)
                             $'.2*chop            Remove last digit from
                                                  the top row, multiply by 2
                                                  and append to bottom row
                                       $&/2 .$_   Divide removed digit by
                                                  2 and prepend it to the
                                                  top row
                          $_=         |           "or" the top and bottom
                                                  row together. The ASCII
                                                  values of 0,1,2,3 have
                                                  00,01,10,11 as their last
                                                  two bits.
y;0-3; '.:                  Convert the smashed together top and bottom rows
                            to the corresponding representation characters.
                            Drop the final ; since it is provided by -p
                            (after a newline which doesn't matter here)
```
Space  is not converted in the above code. For the calculations `/2` and `*2` it will behave like and become `0`. In the other positions it will be part of the "or", but the 1 bits of space are a subset of the one bits of `0` and will have the same effect as `0` if or-ed with any of the digits. Only if the character it is or-ed with is a space will it remain a space instead of becoming a `0`. But that's ok since `0` would have been converted back to space anyways.
[Answer]
# Jelly, ~~32~~ ~~30~~ 29 bytes
```
,Ṛe€"“':“.:”µṪ€;"ṚU1¦ZḄị“'.: 
```
Note the trailing space. [Try it online!](http://jelly.tryitonline.net/#code=LOG5mmXigqwi4oCcJzrigJwuOuKAncK14bmq4oKsOyLhuZpVMcKmWuG4hOKAmOG7i-KAnCAnLjo&input=&args=IjouLjonLiI) or [verify all test cases](http://jelly.tryitonline.net/#code=LOG5mmXigqwi4oCcJzrigJwuOuKAncK14bmq4oKsOyLhuZpVMcKmWuG4hOKAmOG7i-KAnCAnLjrigJ0Kw4figqzhub7igqxq4oG3&input=&args=IiAgIiwiIC4iLCIuICIsIicgIiwiICciLCI6ICIsIicnIiwiOi4iLCI6LiciLCIuLi4iLCI6IDoiLCInOi4iLCIuOiciLCIgICAgIiwiOjo6OiIsIjouLjonLiIsIiA6ICAuOiciLCI6ICcnLiA6IiwiLi4uLi4uLi4iLCI6Ojo6ICAgICIsIiAgICA6Ojo6IiwiIDogOiA6IDogIiwiLicuJy4nLicuJyIsIi4uIDo6ICAnLicgOi4iLCIuJyAgJy46OiAgOi4nLiAuIDo6LicgICcuIC4gLjo6JyAgOi4nLiI).
### Background
We begin by considering the input string (e.g., `:..:'.`) and its reverse.
```
:..:'.
.':..:
```
For each character in the top row, we check if it belongs to `':`, and for each character of the bottom row if it belongs to `.:`. This gives the 2D array of Booleans
```
100110
101111
```
which is the matrix from the question, with reversed bottom row.
We remove the last Boolean of each row, reverse the order of the rows, prepend the Booleans in their original order, and finally reverse the top row.
```
100110    10011    10111    010111    111010
101111    10111    10011    110011    110011
```
This yields the rotated matrix from the question.
Finally, we consider each column of Booleans a binary number and index into `'.:`  to obtain the appropriate characters.
```
332031    ::. :'
```
### How it works
```
,Ṛe€"“':“.:”µṪ€;"ṚU1¦ZḄ‘ị“'.:   Main link. Argument: S (string)
 Ṛ                              Reverse S.
,                               Form a pair of S and S reversed.
     “':“.:”                    Yield ["':" ".:"].
  e€"                           For each character in S / S reversed, check if it
                                is an element of "':" / ".:".
                                Yield the corresponding 2D array of Booleans.
            µ                   Begin a new, monadic chain.
                                Argument: A (2D array of Booleans)
             Ṫ€                 Pop the last Boolean of each list.
                 Ṛ              Yield the reversed array of popped list.
               ;"               Prepend the popped items to the popped lists.
                  U1¦           Reverse the first list.
                     Z          Zip to turn top and bottom rows into pairs.
                      Ḅ         Convert each pair from base 2 to integer.
                        “'.:    Yield "'.: ".
                       ị        Retrieve the characters at the corr. indices.
```
[Answer]
# [Retina](https://github.com/mbuettner/retina), 66
* 2 bytes saved thanks to @daavko
* 4 bytes saved thanks to @randomra
```
:
1e
\.
1f
'
0e
 
0f
T`h`Rh`^.|.$
(.)(\d)
$2$1
e1
:
e0
'
f0
 
f1
.
```
### Explanation
Starting with input:
```
: ''. :
```
The first 4 stages construct the matrix, using `1`/`e` for true and `0`/`f` for false for the top/bottom rows, respectively. Top and bottom rows are interlaced together. This would yield a string like:
```
e1f0e0e0f1f0e1
```
However, these 4 stages also effectively move the lower row 1 to the left, simply by reversing the order of the letters and digits:
```
1e0f0e0e1f0f1e
```
The `T`ransliteration stage reverses hex digits for the first and last characters only, i.e. replaces `0-9a-f` with `f-a9-0`. This has the effect of moving the bottom-left character up to the top row and the top-right character down to the bottom row:
```
ee0f0e0e1f0f11
```
The next stage then swaps every letter-digit pair, thereby moving the upper row 1 to the right. Previously this was `(\D)(\d)`, but it turns out that `(.)(\d)` is sufficient because the substitutions always happen left-to-right and so the final two digits won't be erroneously matched by this, because penultimate character will have been already substituted. The matrix has now been fully rotated as required:
```
e0e0f0e1e0f1f1
```
The final 4 stages then translate back to the original format:
```
'' :'..
```
[Try it online.](http://retina.tryitonline.net/#code=OgoxZQpcLgoxZgonCjBlCiAKMGYKVGBoYFJoYF4ufC4kCiguKShcZCkKJDIkMQplMQo6CmUwCicKZjAKIApmMQou&input=OiAnJy4gOg)
[All testcases, one per line](http://retina.tryitonline.net/#code=OgoxZQpcLgoxZgonCjBlCiAKMGYKVG1gaGBSaGBeLnwuJAooLikoXGQpCiQyJDEKZTEKOgplMAonCmYwCiAKZjEKLg&input=ICAKIC4KLiAKJyAKICcKOiAKJycKOi4KOi4nCi4uLgo6IDoKJzouCi46JwogICAgCjo6OjoKOi4uOicuCiA6ICAuOicKOiAnJy4gOgouLi4uLi4uLgo6Ojo6ICAgIAogICAgOjo6OgogOiA6IDogOiAKLicuJy4nLicuJwouLiA6OiAgJy4nIDouCi4nICAnLjo6ICA6LicuIC4gOjouJyAgJy4gLiAuOjonICA6Licu), `m` added to `T` line to allow separate treatment of each input line.
[Answer]
# Python 3, ~~145~~ ~~141~~ 130 bytes
```
def f(s):a=[i in"':"for i in s]+[i in".:"for i in s][::-1];return''.join(" '.:"[i+2*j]for i,j in zip([a[-1]]+a,a[-2:len(s)-2:-1]))
```
## Explanation
The golfed solution use the following property of zip : `zip('ABCD', 'xy') --> Ax By` 
so `zip(a[:l],a[l:])`  can be replace by `zip(a,a[l:])` and that allow to remove the definition of `l`
```
def f(s):
 l=len(s)-1
 #                ┌───── unfold input string :  123  -> 123456
 #                │                             654
 #  ──────────────┴──────────────────────────────
 a=[i in"':"for i in s]+[i in".:"for i in s][::-1]
 # ─────────┬─────────   ────────────┬───────────
 #          │                        └──── generate the second row and reverse it
 #          └─────────── generate the first row 
 return''.join(" '.:"[i+2*j]for i,j in zip([a[-1]]+a[:l],a[l:-1][::-1]))
 #             ──────┬──────           ─┬    ────────────┬───────────
 #                   │                  │                └──── rotate and create first/second new row :  123456  -> 612345  -> 612
 #                   │                  │                                                                                      543
 #                   │                  └ group pair of the first and second row : 612 -> (6,5),(1,4),(2,3)
 #                   │                                                             543
 #                   └─────────── replace pair by symbol 
```
## Results
```
>>> f(".'  '.::  :.'. . ::.'  '. . .::'  :.'.")
"' ' .:.''..'.'. ..:' ' .'. ...'''..'.'"
>>> f(".....''''''")
":...  '''':"
```
[Answer]
# Pyth, ~~38~~ 36
```
L,hb_ebsXCyc2.>syCXzJ" .':"K.DR2T1KJ
```
2 bytes thanks to Jakube!
[Try it here](http://pyth.herokuapp.com/?code=L%2Chb_ebsXCyc2.%3EsyCXzJ%22+.%27%3A%22K.DR2T1KJ&input=..+%3A%3A++%27.%27+%3A.&debug=0) or run the [Test Suite](http://pyth.herokuapp.com/?code=L%2Chb_ebsXCyc2.%3EsyCXzJ%22+.%27%3A%22K.DR2T1KJ&test_suite=1&test_suite_input=++%0A+.%0A.+%0A%27+%0A+%27%0A%3A+%0A%27%27%0A%3A.%0A%3A.%27%0A...%0A%3A+%3A%0A%27%3A.%0A.%3A%27%0A++++%0A%3A%3A%3A%3A%0A%3A..%3A%27.%0A+%3A++.%3A%27%0A%3A+%27%27.+%3A%0A........%0A%3A%3A%3A%3A++++%0A++++%3A%3A%3A%3A%0A+%3A+%3A+%3A+%3A+%0A.%27.%27.%27.%27.%27%0A..+%3A%3A++%27.%27+%3A.%0A.%27++%27.%3A%3A++%3A.%27.+.+%3A%3A.%27++%27.+.+.%3A%3A%27++%3A.%27.&debug=0).
### Explanation:
```
L,hb_eb         ##  Redefine the function y to take two lists
                ##  and return them but with the second one reversed
                ##  Uses W to apply a function only if it's first argument is truthy
XzJ" .':"K.DR2T ##  Does a translation from the string " .':" to
                ##  .DR2T which is [0,1,2,3...,9] mapped to divmod by 2
                ##  (which is [0,0],[0,1],[1,0],[1,1], then some extra, unused values)
                ##  we also store the string and the list for later use in J and K
.>syC ... 1     ##  zip the lists to get the bits on top and below as two separate lists
                ##  apply the function y from before, flatten and rotate right by 1
Cyc2            ##  split the list into 2 equal parts again, then apply y and zip again
sX ... KJ       ##  apply the list to string transformation from above but in reverse
                ##  then flatten into a string
```
[Answer]
# Pyth, 66 bytes
```
KlQJ.nCm@[,1Z,Z1,ZZ,1 1)%Cd5Qjkm@" .':"id2Ccs[:JKhK<JtK>JhK:JtKK)K
```
[Try it here!](http://pyth.herokuapp.com/?code=KlQJ.nCm%40%5B%2C1Z%2CZ1%2CZZ%2C1%201%29%25Cd5Qjkm%40%22%20.%27%3A%22id2Ccs%5B%3AJKhK%3CJtK%3EJhK%3AJtKK%29K&input=%22%20.%22&test_suite=1&test_suite_input=%22%20%20%22%0A%22%20.%22%0A%22.%20%22%0A%22%27%20%22%0A%22%20%27%22%0A%22%3A%20%22%0A%22%27%27%22%0A%22%3A.%22%0A%22%3A.%27%22%0A%22...%22%0A%22%3A%20%3A%22%0A%22%27%3A.%22%0A%22.%3A%27%22%0A%22%20%20%20%20%22%0A%22%3A%3A%3A%3A%22%0A%22%3A..%3A%27.%22%0A%22%20%3A%20%20.%3A%27%22%0A%22%3A%20%27%27.%20%3A%22%0A%22........%22%0A%22%3A%3A%3A%3A%20%20%20%20%22%0A%22%20%20%20%20%3A%3A%3A%3A%22%0A%22%20%3A%20%3A%20%3A%20%3A%20%22%0A%22.%27.%27.%27.%27.%27%22%0A%22..%20%3A%3A%20%20%27.%27%20%3A.%22%0A%22.%27%20%20%27.%3A%3A%20%20%3A.%27.%20.%20%3A%3A.%27%20%20%27.%20.%20.%3A%3A%27%20%20%3A.%27.%22&debug=0)
## Explanation
This can be broken down in 3 parts:
* Convert the input into a flat array of ones and zeros.
* Do the rotation.
* Convert it back into ASCII.
### Convert input
This is fairly trivial. Each character is mapped in the following way:
```
  -> (0,0)
. -> (0,1)
' -> (1,0)
: -> (1,0)
```
*First one is a whitespace.*  
We get a list of 2-tuples which we transpose to get the 2 rows of the matrix which then gets flattened.
**Code**
```
KlQJ.nCm@[,1Z,Z1,ZZ,1 1)%Cd5Q   # Q = input
KlQ                             # save the width of the matrix in K (gets used later)
       m                    Q   # map each character d
                        %Cd5    # ASCII-code of d modulo 5
        @[,1Z,Z1,ZZ,1 1)        # use this as index into a lookup list
   J.nC                         # transpose, flatten and assign to J
                        
```
### Rotate
We have the matrix as flat array in `J` and the width of the matrix in `K`. The rotation can be described as:
```
J[K] + J[:K-1] + J[K+1:] + J[K-1]
```
**Code**
```
s[:JKhKJhK:JtKK)    # J = flat array, K = width of matrix
s[                  )    # Concat all results in this list
  :JKhK                  # J[K]
       JhK          # J[K+1:]
               :JtKK     # J[K-1]
```
## Convert it back
```
jkm@" .':"id2Cc[)K    # [) = resulting list of the step above
              c[)K    # chop into 2 rows
             C        # transpose to get the 2-tuples back
  m                   # map each 2-tuple d
          id2         # interpret d as binary and convert to decimal
   @" .':"            # use that as index into a lookup string to get the correct char
jk                    # join into one string
```
[Answer]
# Python 3, ~~166~~ ~~154~~ ~~153~~ ~~150~~ ~~146~~ ~~138~~ ~~137~~ ~~135~~ ~~132~~ 127 bytes
**Edit:** I've borrowed the use of `zip` from [Erwan's Python answer](https://codegolf.stackexchange.com/a/75353/47581) at the end of the function. ~~and their idea to use `[::-1]` reversals, though I put in my own twist.~~ As it turns out, reversals were not a good idea for my function. I have changed my use of `format` for further golfing. Moved `a` and `b` directly into `zip` for further golfing (ungolfing remains unchanged because the separation of `a` and `b` there is useful for in avoiding clutter in my explanation)
**Edit:** Borrowed `(some number)>>(n)&(2**something-1)` from [this answer by xnor on the Music Interval Solver challenge](https://codegolf.stackexchange.com/a/76296/47581). The clutter that is `zip(*[divmod(et cetera, 2) for i in input()])` can probably be golfed better, though I do like the expediency it grants from using two tuples `t` and `v`.
```
t,v=zip(*[divmod(708>>2*(ord(i)%5)&3,2)for i in input()])
print("".join(" '.:"[i+j*2]for i,j in zip((v[0],*t),(*v[1:],t[-1]))))
```
**Ungolfed:**
```
def rotate_dots(s):
    # dots to 2 by len(s) matrix of 0s and 1s (but transposed)
    t = []
    v = []
    for i in s:
        m = divmod(708 >> 2*(ord(i)%5) & 3, 2)
            # ord(i)%5 of each char in . :' is in range(1,5)
            # so 708>>2 * ord & 3 puts all length-2 01-strings as a number in range(0,4)
            # e.g. ord(":") % 5 == 58 % 5 == 3
            # 708 >> 2*3 & 3 == 0b1011000100 >> 6 & 3 == 0b1011 == 11
            # divmod(11 & 3, 2) == divmod(3, 2) == (1, 1)
            # so, ":" -> (1, 1)
        t.append(m[0])
        v.append(m[1])
    # transposing the matrix and doing the rotations
    a = (v[0], *t)          # a tuple of the first char of the second row 
                            # and every char of the first row except the last char
    b = (v[1:], t[-1])      # and a tuple of every char of the second row except the first
                            # and the last char of the first row
    # matrix to dots
    z = ""
    for i, j in zip(a, b):
        z += " '.:"[i + j*2]    # since the dots are binary
                                # we take " '.:"[their binary value]
    return z
```
[Answer]
# [MATL](https://esolangs.org/wiki/MATL), ~~40~~ 39 bytes
```
' ''.:'tjw4#mqBGnXKq:QKEh1Kq:K+hv!)XBQ)
```
[**Try it online!**](https://tio.run/##y00syfn/X11BXV3PSr0kq9xEObfQyT0vwrvQKtDbNcMQSHtrZ6iVKWpGOAVq/v@vp66gAFRqpaBgpaeup6CnYGUFEQIygcLqEHEA) *The linked version has `v` repaced by `&v`, because of changes in the language after this answer was posted*.
```
' ''.:'               % pattern string. Will indexed into, twice: first for reading 
                      % the input and then for generating the ouput
t                     % duplicate this string
j                     % input string
w                     % swap
4#m                   % index of ocurrences of input chars in the pattern string
qB                    % subtract 1 and convert to binay. Gives 2-row logical array
GnXKq:QKEh1Kq:K+hv!   % (painfully) build two-column index for rotation
)                     % index into logical array to perform the rotation
XBQ                   % transform each row into 1, 2, 3 or 4
)                     % index into patter string. Implicitly display
```
[Answer]
## Ruby, ~~166~~ 163 bytes
```
->s{a=s.tr(f=" .':",t='0-3').chars.map{|x|sprintf('%02b',x).chars}.transpose;a[1]+=[a[0].pop];a[0]=[a[1].shift]+a[0];a.transpose.map{|x|x.join.to_i 2}.join.tr t,f}
```
Yuck... `transpose` is too long.
Tricks used here:
* `sprintf('%02b',x)` to convert `"0"`, `"1"`, `"2"`, `"3"` into `"00"`, `"01"`, `"10"`, and `"11"` respectively. Surprisingly, the second argument does *not* have to be converted into an integer first.
* The rotation is done via `a[1].push a[0].pop;a[0].unshift a[1].shift;`, which I thought was at least a little clever (if not excessively verbose in Ruby). The symmetry is aesthetically nice, anyway :P
[Answer]
# Javascript ES6 125 bytes
```
q=>(n=[...q].map(a=>(S=` .':`).indexOf(a))).map((a,i)=>(i?n[i-1]&2:n[0]&1&&2)|((I=n[i+1])>-1?I&1:n[i]&2&&1)).map(a=>S[a]).join``
```
---
I map each character to a two digit binary equivalent
```
 : becomes 3   11
 ' becomes 2   10
 . becomes 1   01
   becomes 0   00
```
and I'm thinking of them as being one on top of the other
```
3212021 becomes
1101010
1010001
```
I save that to n
For each character (0-3) of n, I check it's neighbors, adding the highest order bit of the left neighbor to the lowest order bit of the right neighbor.
if i==0 (first character) I use it's own lower order bit instead of the left neighbor's higher order bit.
if n[i+1]>-1 it means we got 0,1,2,3 so when that's false we hit the last element.
When that happens I use the character's own highest order bit instead of the right neighbor's lower bit
map that back to  `.':` land and join that array back together
[Answer]
# [Common Lisp](http://www.clisp.org/), ~~253~~ 247 bytes
```
(defun R(i &aux(n(length i))(u #1=(make-list n))(b #1#))(dotimes(k n)(setf(elt u k)(find #2=(elt i k)"':")(elt b k)(find #2#".:")))(map'string(lambda(u b)(cond((and u b)#\:)(u #\')(b #\.)(t #\ )))`(,(car b),@(butlast u))`(,@(cdr b),(nth(1- n)u))))
```
[Try it online!](https://tio.run/##dVTBattAED3XXzHIpDsy8UIMvagEUkIKgZ5seqoLXUvrRFiSVWlV0ot@3X2zsqWNoWuwZt68ffNmrXVa5G19OnFm911Fa87po@neuOLCVi/ulfI45o7md/dcmoNdgu2oArYDNsczO7q8tC0fgHJr3Z5t4aijQ8z7vMpovrr3SA4kUkkU@2wX1OeRBgyp0tSqdU1evXBhyl1m0HgXc3qsMmYDrqTzbeINbZX3sNUxOzwIAr/4llPTgHT7wLvOFQZeO48/cJr5Alfule@WMItCHJ9mMnhtGlNaZxtaONu6ZWpa2y5mRIo5IopIvmLkRMg1cj3lWupqypXnqzFPfH3KEQJJprroJQFfKw/oqYP2lKBngv0QTSaO8jI60JGYPHN0TudZgmkSLKHJY3Kgpb2gmgJFSogGVU1K63AGUkoPnhR26Hfe/fID@PW@99kTIuwMTxnr7A0RDAX@4GP4eCdDGMytLh@xM2aBI5K2wGg4eqijt09DFaEIEagm2TNACAGrAY88W35@CADA5MrTcUwC@RDwgMtLPpudb9r4ohGPoTTHvXMLYs6runOEO9LgJZ4Y8ewDsX2rbepsRrhwcjuu641tO1wyXpNXkbZ@qv2xKQ3UKHoWPOlXn1zf9jdPF7213zjiV2mf/Pj65flb/3nz/fHxabPpf/armwgNsQa7o7GLA/u7M0V9jV/8/K8qp1Tjf0AE/tjmL83VdFzhFQXz9A8 "Common Lisp – Try It Online")
Tests included in TIO.
```
(defun R (i &aux (n (length i))  ; take the length of input
                 (u #1=(make-list n))   ; create the top (up) list
                 (b #1#))  ; create the bottom list
  (dotimes (k.. n)
    (setf (elt u k) (find #2=(elt i k) "':") 
; set the k'th element of the top list to a non-falsey value if k'th element of input is #\' or #\:
          (elt b k) (find #2# ".:")
; set the k'th element of the bottom list to a non-falsey value if k'th element of input is #\. or #\:
(map 'string  ; collect the results into a string
     (lambda (u b) ; for each element of the rotated up and bottom lists
       (cond
        ((and u b) #\:) ; if both top and bottom are non-falsey add #\:
        (u #\')         ; if only top is non-falsey add #\'
        (b #\.)         ; if only bottom is non-falsey add #\.
        (t #\ )))       ; if both are falsey add '#\ ' (#\Space)
     `(,(car b),@(butlast u))   ; first element of the bottom and the all but last element of the top
     `(,(cdr b),(nth(1- n)u)))) ; all but first element of the bottom and the last element of the top
```
*Saved 6 bytes using* `#=` *notation for* `(make-list n)` *and* `(elt i k)`
[Answer]
# JavaScript, 311 bytes
Can probably be improved alot:
```
a=(s=prompt()).length-1;o=s[0]==":"||s[0]=="."?s[1]==":"||s[1]=="."?":":"'":s[1]==":"||s[1]=="."?".":" ";for(i=1;i<a;i++)o+=s[i-1]=="'"||s[i-1]==":"?s[i+1]=="."||s[i+1]==":"?":":"'":s[i+1]=="."||s[i+1]==":"?".":" ";alert(o+=s[a]==":"||s[a]=="'"?s[a-1]==":"||s[a-1]=="'"?":":"'":s[a-1]==":"||s[a-1]=="'"?".":" ")
```
[Answer]
## JavaScript (ES6), 237 210 204 188 182 178 bytes
*Credit to [@Downgoat](https://codegolf.stackexchange.com/users/40695/downgoat) for saving 16 bytes in the 188-byte revision*
*Update: I had a brainwave and reduced the first operation on `s` to a single `map` call instead of two separate ones*
```
s=>(r=" .':",a=[],s=[...s].map(c=>(t=('00'+r.indexOf(c).toString(2)).slice(-2),a.push(t[0]),t[1])),a.splice(0,0,s.shift()),s.push(a.pop()),a.map((v,i)=>r[+('0b'+v+s[i])]).join``)
```
### Pretty Print & Explanation
```
s => (
  r = " .':", // Map of characters to their (numerical) binary representations (e.g. r[0b10] = "'")
  a = [],     // extra array needed
  // Spread `s` into an array
  s = [...s].map(c => (
    // Map each character to a `0`-padded string representation of a binary number, storing in `t`
    t = ('00' + r.indexOf(c).toString(2)).slice(-2)),
    // Put the first character of `t` into `a`
    a.push(t[0]),
    // Keep the second character for `s`
    t[1]
  )),
  // Put the first character of `s` in the first index of `a`
  a.splice(0,0,s.shift()),
  // Append the last character of `a` to `s`
  s.push(a.pop(),
  // Rejoin the characters, alternating from `a` to `s`, representing the rotated matrix, and map them back to their string representation
  // Use implicit conversion of a binary number string using +'0b<num>'
  a.map((v,i) => r[+('0b' + v + s[i])]).join``
)
```
[Answer]
## Perl, ~~144~~ ~~142~~ ~~137~~ 131 bytes
```
y/.':/1-3/;s/./sprintf'%02b ',$&/ge;@a=/\b\d/g;@b=(/\d\b/g,pop@a);@a=(shift@b,@a);say map{substr" .':",oct"0b$a[$_]$b[$_]",1}0..@a
```
Byte added for the `-n` flag.
Pretty much the same algorithm as [my Ruby answer](https://codegolf.stackexchange.com/a/75303/3808), just shorter, because... Perl.
```
y/.':/1-3/;                         # transliterate [ .':] to [0123]
s/./sprintf'%02b ',$&/ge;           # convert each digit to 2-digit binary
@a=/\b\d/g;                         # grab the 1st digit of each pair
@b=(/\d\b/g,                        # 2nd digit of each pair
pop@a);                             # push the last element of a to b
@a=(shift@b,@a);                    # unshift the first element of b to a
say                                 # output...
map{                                # map over indices of a/b
substr" .':",oct"0b$a[$_]$b[$_]",1  # convert back from binary, find right char
}0..@a                              # @a is length of a
```
Obnoxiously, `@a=(shift@b,@a)` is shorter than `unshift@a,shift@b`.
Alas, these are the same length:
```
y/ .':/0-3/;s/./sprintf'%02b ',$&/ge;
s/./sprintf'%02b ',index" .':",$&/ge;
```
Thanks to [Ton Hospel](https://codegolf.stackexchange.com/users/51507/ton-hospel) for 5 bytes and [msh210](https://codegolf.stackexchange.com/users/1976/msh210) for a byte!
[Answer]
# [Vyxal](https://github.com/Lyxal/Vyxal), 44 bytes
```
f`:'. `f⁺↑b2ẇĿ2ẇÞT÷ḣ„ṫ$„p‟JWÞT⁺↑b2ẇ`:'. `fĿ∑
```
[Try it Online!](https://lyxal.pythonanywhere.com?flags=&code=f%60%3A%27.%20%60f%E2%81%BA%E2%86%91b2%E1%BA%87%C4%BF2%E1%BA%87%C3%9ET%C3%B7%E1%B8%A3%E2%80%9E%E1%B9%AB%24%E2%80%9Ep%E2%80%9FJW%C3%9ET%E2%81%BA%E2%86%91b2%E1%BA%87%60%3A%27.%20%60f%C4%BF%E2%88%91&inputs=%27%3A..%3A%5C%27.%27&header=&footer=)
Yuck.
[Answer]
# [Python 3](https://docs.python.org/3/), 124 bytes
```
def f(a,s=" .':",j="".join):q=j(bin(4+s.find(i))for i in a);return j(s[int(j(x),2)]for x in zip(q[4]+q[3::5],q[9::5]+q[-2]))
```
[Try it online!](https://tio.run/##TVK7bsMwDJybryC0UEJUDkk6RIW/xPCQIjYqD05sq0Dan3dJUUorAyHv@Lizo/t3@rxNx2279gMM9uLXxgBhMH5sjKHxFicX5ma0H3Gyp/1KQ5yuNjo33BaIECe4uPelT1/LBKNd2zglO9qH8wfXSctDWn7i3c7tqdvP7TGEt87P7Vki49dD59yW@jWtAE27a1ndeDAEpvOMODLCgjAjQEUcBZGioJ2lhloLpZZ3BqwIFZZJIi1XzcBjsiqUOuo41XlJpL9uBz5ZTaJu4JNbJBZNEr3MEdRNEADKNgIkevoFRCoukLvpz2k@ajeff4rVB6c8Vb0IW/1wyj6qJ1bXR/U1r@@I9ckmnrD6AJFjAsqn5b2smfFzgzRIG1MEMqEUp0yj8vrv8iQDfk3MrfxJhMop08qbrtvJjUq@lzuVr0zYvdwXuXKDTc6D/ELTQO@2Xw "Python 3 – Try It Online")
## How it works :
* `q="".join(bin(4+s.find(i))for i in a)` store an unpacked version of our starting string `a` by using binary conversion
If `a` is equal to
```
ABCD
EFGH
```
then `q` is equal to `---ae---bf---cg---dh` with a to h equal to `"1"` if there is a dot in the corresponding emplacement and `"0"` otherwise (and `---` equal to `0b1`)
* `zip(q[4]+q[3::5],q[9::5]+q[-2])` reshapes `q` to have : `(eabcd),(fghd)` (the last `d` in the first part is ignored since the second part is shorter)
* `s[int("".join(x),2)]` reverse the process converting back the binary number into their dot version. Result :
```
EABC
FGHD
```
[Answer]
# [05AB1E](https://github.com/Adriandmen/05AB1E), 41 bytes
```
Â)εS„':„.:‚Nèså¤ˆ¨}R¯øí˜2äćR¸ìζJC" '.:"sè
```
[Try it online!](https://tio.run/##yy9OTMpM/f//cJPmua3BjxrmqVsBCT0gMcvv8Iriw0sPLTnddmhFbdCh9Yd3HF57eo7R4SVH2oMO7Ti85tw2L2clBXU9K6Xiwyv@e/3XU1cA8awUFKz01PUU9BSsrCBCQCZQWB0iDgA "05AB1E – Try It Online") Outputs as a list of characters. Link includes a footer to format the output.
```
Â)εS„':„.:‚Nèså¤ˆ¨}R¯øí˜2äćR¸ìζJC"..."sè  # trimmed program
                                       è  # push character...
                                          # (implicit) s...
                                       è  # in...
                                 "..."s   # literal...
                                       è  # at...
                                      s   # (implicit) indices in...
                              ζ           # list of elements with same indices in each element of...
                            ¸             # list of...
                          ć               # first element of...
 )                                        # list of...
Â                                         # implicit input...
 )                                        # and...
                                          # implicit input...
Â                                         # reversed...
  ε                                       # with each element replaced by...
              å¤                          # is...
             s                            # (implicit) each element of...
   S                                      # list of characters of...
                                          # (implicit) current element in map...
              å                           # in...
            ès                            # element in...
          ‚                               # list of...
    „':                                   # literal...
          ‚                               # and...
       „.:                                # literal...
            è                             # at index...
           N                              # current index in map...
                 ¨                        # excluding the last element...
                ˆ                         # and append...
               ¤                          # last element of...
              å¤                          # is...
             s                            # (implicit) each element of...
   S                                      # list of characters of...
                                          # (implicit) current element in map...
              å                           # in...
            ès                            # element in...
          ‚                               # list of...
    „':                                   # literal...
          ‚                               # and...
       „.:                                # literal...
            è                             # at index...
           N                              # current index in map...
                ˆ                         # to global array...
                   R                      # reversed...
                      í                   # with...
                    ¯                     # global array...
                     ø                    # with each element paired with corresponding element in...
                    ¯                     # global array...
                      í                   # prepended...
                       ˜                  # flattened...
                         ä                # split into...
                        2                 # literal...
                         ä                # equal pieces...
                           R              # reversed...
                             ì            # with...
 )                                        # list of...
Â                                         # implicit input...
 )                                        # and...
                                          # implicit input...
Â                                         # reversed...
  ε                                       # with each element replaced by...
              å¤                          # is...
             s                            # (implicit) each element of...
   S                                      # list of characters of...
                                          # (implicit) current element in map...
              å                           # in...
            ès                            # element in...
          ‚                               # list of...
    „':                                   # literal...
          ‚                               # and...
       „.:                                # literal...
            è                             # at index...
           N                              # current index in map...
                 ¨                        # excluding the last element...
                ˆ                         # and append...
               ¤                          # last element of...
              å¤                          # is...
             s                            # (implicit) each element of...
   S                                      # list of characters of...
                                          # (implicit) current element in map...
              å                           # in...
            ès                            # element in...
          ‚                               # list of...
    „':                                   # literal...
          ‚                               # and...
       „.:                                # literal...
            è                             # at index...
           N                              # current index in map...
                ˆ                         # to global array...
                   R                      # reversed...
                      í                   # with...
                    ¯                     # global array...
                     ø                    # with each element paired with corresponding element in...
                    ¯                     # global array...
                      í                   # prepended...
                       ˜                  # flattened...
                         ä                # split into...
                        2                 # literal...
                         ä                # equal pieces...
                          ć               # excluding the first element...
                             ì            # prepended...
                                          # (implicit) with each element...
                               J          # joined...
                                          # (implicit) with each element...
                                C         # converted from binary to decimal
```
[Answer]
# [Python 3](https://docs.python.org/3/), ~~105~~ 104 bytes
```
s=" .':"
a=[*map(s.find,input())]
b=a[0]*2,*a,a[-1]//2
print(*map(lambda x,y:s[x&2|y&1],b,b[2:]),sep='')
```
[Try it online!](https://tio.run/##HctLCsMgEADQvaeQLOKHqWnsbsCTiIuRtFRorEQLEXp3C337V3p7vvNtjOombgROjJzXOxVZzSPlDVIunyaVCiw68tegLWgC8pc1LItl5Ui5yX940R434id0rP6c7bfPa4AI0VsMCuq9OCHUGByNQWG44QLHDw "Python 3 – Try It Online")
[Answer]
# Python 3, ~~294~~ ~~287~~ 283 bytes
Waaayyyyyy too long, but I will try to golf of some bytes:
```
z=input()
x=len(z)
M=[0,1,2,3]
for Q in M:z=z.replace(":'. "[Q],"11100100"[Q*2:Q*2+2])
a=[]
b=[]
for X in range(x):a+=[z[X*2]];b+=[z[X*2+1]]
b=b[1:]+[a.pop()]
c=[b[0]]+a
z=""
for X in range(len(c)):
 y=c[X]+b[X]
 for Q in M:y=y.replace("11100100"[Q*2:Q*2+2],":'. "[Q])
 z+=y
print(z)
```
[Answer]
## Lua, 139 bytes
```
print(((...):gsub(".",{[" "]="NN@",["."]="YN@",["'"]="NY@",[":"]="YY@"}):gsub("(.)@(.?)","%2%1"):gsub("..",{NN=" ",NY=".",YN="'",YY=":"})))
```
Usage: 
```
$ lua conveyor.lua ".'  '.::  :.'. . ::.'  '. . .::'  :.'."
' ' .:.''..'.'. ..:' ' .'. ...'''..'.'
```
[Answer]
# Excel, 139 bytes
```
=LET(e,LEN(A1),q,SEQUENCE(e),x,CODE(MID(A1,q,1))-32,l,x>7,u,x>l*14,CONCAT(MID(" '.:",IF(q=1,l,INDEX(u,q-1))+IF(q=e,u,INDEX(l,q+1))*2+1,1)))
```
[Link to Spreadsheet](https://1drv.ms/x/s!AkqhtBc2XEHJnD_4qim2waVo0S2j?e=ghmLxp)
## How it works
```
=LET(
*Assignments*
e,LEN(A1),              'e = length of A1
q,SEQUENCE(e),          'q = (1 .. e)
x,CODE(MID(A1,q,1))-32, 'x = list of ASCII codes of each character - 32
l,x>7,                  'l = array of lower numbers = 1 if x>7 (.:)
u,x>l*14,               'u = array of upper numbers = 1 if x>l*14 (':)
*Result*
IF(q=1,l,INDEX(u,q-1)) +   'String indices = if q = 1 then l[1] else u[q-1] 
IF(q=e,u,INDEX(l,q+1))*2+1 '               + (if q=e then u[e] else l[q+1]) * 2 + 1
CONCAT(MID(" '.:",~,1)))   'Concatenate all the results
```
[Answer]
# [J](http://jsoftware.com/), 55 bytes
```
' .'':'&([{~([:g@($$_1|.,)g=.|.@{:1}_2{.])&.(|:@#:)@i.)
```
[Try it online!](https://tio.run/##VVJBasMwELz7FUsbIguSoelxIWAI9JRTriHkEJq0vfQBdvt1Z3dtadeyDNJoNDs79s/4gnSnPVOiDb0Ry7sFHU7HjzERUuK0bs/9f3vmR9euVtfdgE1@7DGg63n3d33vcclrtAN3r5y7b@QxN83n7etXFGVume62mjE4hoKlVEFUosETqOcFRb0ucL3vXHYq@33ncqAigAiok90tEOyys6VK6Yyj4yDuRoxSw4nxeEAso6rruuKghXmYa49VLHol@ZAUq1lCeiE4VwzuXxR1xI5tBGOm4rbVXrQuS3Hl9vUstiDF7PEuZsA/mrYUZ/GyxD1awDxBu4PrahRizcjEix9NyMbUKEyOLEmaVHSH6YeZz4MFE1RdnphSo6CygeYzn6VmfAI "J – Try It Online")
]