Unscramble those case-(very)-sensitive strings

Retina, 53 bytes

Not really clever, but a clean and quite readable solution

([a-z])(.*?)([a-z])
$3$2$1
([A-Z])(.*?)([A-Z])
$3$2$1

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MATL, 22 bytes

2:"tttk<f2etAZ))P5M(Yo

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How it works

2:"       % Do the following twice
  ttt     %   Input string (implicit). Push three more copies
  k       %   Convert to lowercase
  <f      %   Indices of characters that had their code point increased by
          %   the lowercase conversion, i.e. that were uppercase letters
  2e      %   Convert to 2-row matrix. This pads a zero in the lower-right 
          %   corner if necessary
  tAZ)    %   Keep only columns that don't contain zeros. Thus if there
          %   was a character that can't be swapped it will be ignored             
  )       %   Get 2-row matrix of characters at those positions
  P       %   Flip vertically. This does the swapping
  5M      %   Push matrix of original indices again
  (       %   Write the swapped characters onto their original positions
  Yo      %   Change case. In the first iteration, this prepares the
          %   string so the second iteration will process the letters that
          %   were originally lowercase. In the second iteration, it
          %   undoes the change of case 
          % End (implicit)
          % Display (implicit)

Jelly, 21 20 19 18 bytes

s2UF,
nŒlTÇyJịŒsµ⁺

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How it works

nŒlTÇyJịŒsµ⁺  Main link. Argument: s (string)

 Œl           Convert to lowercase.
n             Test for inequality.
   T          Truth; yield all indices of 1's.
    Ç         Call the helper link. Yields [A, B] (pair of lists).
      J       Indices; yield I := [1, ..., len(s)].
     y        Translate; replace the integers of I that occur in A with the
              corresponding integers in B.
        Œs    Swapcase; yield s with swapped case.
       ị      Use the translated index list to index into s with swapped case.
          µ   Combine all links to the left into a chain.
           ⁺   Duplicate the chain, executing it twice.


s2UF,         Helper link. Argument: J (list of indices)

s2            Split J into pairs. If the length is odd, the last list will be
              a singleton list.
  U           Upend; reverse each pair. This is a no-op for singletons lists.
   F          Flatten, concatenating the pairs.
    ,          Pair the previous result with J.