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

Try it online!

MATL, 22 bytes

2:"tttk<f2etAZ))P5M(Yo

Try it online! Or verify all test cases.

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)

Bash + Unix utilities, 77 62 57 56 54 bytes

sed -r "s/([$1)([^$1*)([$1)/\3\2\1/g"||$0 a-z]|$0 A-Z]

Input in stdin. Output in stdout.

(In this last version, stderr happens to be written to also, but PPCG consensus seems to be that that's OK -- stderr is simply ignored.)

Edit 1: Thanks to @Dennis for 15 bytes! Improvements: (a) Taking input via stdin; (b) combining 2 sed scripts into one; and (c) replacing tr with substitution via bash parameter expansion; (b) and (c) disappeared in Edit 2.

Edit 2: Shorter by 5 additional bytes. Used a function call to replace both (b) and (c) in Edit 1.

Edit 3: One more byte -- passed ] as part of the function arguments.

Edit 4: Replaced the two function calls with calls to the program itself when it has no arguments.

Testbed and sample output:

for x in 'lLEhW OroLd!' 'rpGOZmaimgn uplRzse naC DEoO LdGf' 'eIt uqHKC RBWOO xNf ujPMO SzRE HTL EOvd yAg' 'NraWgCi: Nsas-eNEiTIsev rNsiTG!!' '(^_^)' 'AWCTY HUOS RETP' 'hwn oeesd acsp nawyya' 'SpMycaeIesKyBorekn' "D's mroyr, Ivam. I'e faardi I act'n od htta."; do ./swapping <<<"$x" 2>/dev/null; done

hELlO WorLd!
prOGRamming puzZles anD COdE GoLf
tHe quICK BROWN fOx juMPS OvER THE LAzy dOg
WarNiNg: Case-sENsITive sTriNG!!
(^_^)
WATCH YOUR STEP
who needs caps anyway
MySpaceKeyIsBroken
I'm sorry, Dave. I'm afraid I can't do that.

ES6, 185 95 bytes

i=>(o=[...i]).map((c,j)=>/[a-z]/i.test(c)?o[e=c>"Z"]=1/(b=o[e])?o[o[j]=o[b],b]=c:j:0)&&o.join``

Solution severely shortened with the help of @Neil, @Arnauld and @edc65

Explanation

f = i =>
  // Get array of characters from input string
  (o = [...i])
    .map((c, j) => 
      // Check if it's a text character, otherwise skip it
      /[a-z]/i.test(c) ? 
        // Get last character position for case
        // merged with setting a variable for if the character is lowercase
        // merged with storing the current case character position,  
        // under properties on the array (with keys "true" or "false")
        o[e = c>"Z"] =
          // Check if there exists a character position to switch with
          // merged with storing the current position for quick access
          1/(b=o[e]) ? 
            // This statement will end up returning the Array subset, 
            // which will be falsy in the above conditional since (1/[])==false
            o[
              // Switch left character to the right
              o[j]=o[b]
            // Switch right character to the left
            ,b]=c : 
            // No character exists for case, so return current character position
            j
         // It was not a text character, so do nothing
         :0
      )
  // Join array and return as result
  && o.join``;

[
  "lLEhW OroLd!",
  "NraWgCi: Nsas-eNEiTIsev rNsiTG!!",
  "rpGOZmaimgn uplRzse naC DEoO LdGf",
  "eIt uqHKC RBWOO xNf ujPMO SzRE HTL EOvd yAg",
  "(^_^)",
  "AWCTY HUOS RETP",
  "hwn oeesd acsp nawyya",
  "SpMycaeIesKyBorekn",
  "D's mroyr, Ivam. I'e faardi I act'n od htta."
 ].map(testCase => console.log(f(testCase)));

Jelly, 21 20 19 bytes

s2UF,
e€ØATÇyJịŒsµ⁺

Try it online!

How it works

e€ØATÇyJịŒsµ⁺  Main link. Argument: s (string)

eۯA           Test each character in s for membership in "A...Z".
    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.

Python, 82 bytes

lambda s:S(r.lower(),t,S(r,t,s))
import re
S=re.sub
r='([A-Z])(.*?)'*2
t=r'\3\2\1'

Try it online!

Mathematica, 96 bytes

s[#,r="([a-z])(.*?)([a-z])"]~(s=StringReplace[#,RegularExpression@#2->"$3$2$1"]&)~ToUpperCase@r&

A port of Leo's Retina answer, which uses regular expressions.

C++11 (GCC), 154 149 bytes

#include<algorithm>
[](std::string s){int*p,u,l=u=-1;for(auto&c:s)(c|32)-97<26U?p=&(c&32?u:l),~*p?(std::swap(c,s[*p]),*p=-1):*p=&c-&s[0]:0;return s;}

Python 2, 124 bytes

s=input();u=str.isupper
exec"r='';i=0\nfor c in s:r+=c[u(c):]or filter(u,s+s[::-1])[i^1];i+=u(c)\ns=r.swapcase();"*2
print s

Not as short as my regex-based solution, but I think it's still interesting.

Try it online!

Bean, 83 bytes

Hexdump:

00000000 26 53 d0 80 d3 d0 80 a0 5d 20 80 0a a1 81 81 00  &SÐ.ÓÐ. ] ..¡...
00000010 23 81 01 20 80 0a a1 81 81 02 23 81 01 a8 db c1  #.. ..¡...#..¨ÛÁ
00000020 ad da dd a9 a8 db de c1 ad da dd aa bf a9 a8 db  .ÚÝ©¨ÛÞÁ.Úݪ¿©¨Û
00000030 c1 ad da dd 29 a4 b3 a4 b2 a4 31 a8 db e1 ad fa  Á.ÚÝ)¤³¤²¤1¨Ûá.ú
00000040 dd a9 a8 db de e1 ad fa dd aa bf a9 a8 db e1 ad  Ý©¨ÛÞá.úݪ¿©¨Ûá.
00000050 fa dd 29                                         úÝ)
00000053

Equivalent JavaScript:

a.replace(/([A-Z])([^A-Z]*?)([A-Z])/g,'$3$2$1').replace(/([a-z])([^a-z]*?)([a-z])/g,'$3$2$1')

Explanation:

Implicitly taking the first line of input unformatted as a (since newlines can't be part of scrambled string), and implicitly outputs unscrambled string by sequentially replacing uppercase, then lowercase pairs.

Try demo here.

Try test suite here.

JavaScript (ES6), 80 bytes

Based on Leo's Retina answer.

s=>eval("s"+(r=".replace(/([A-Z])([^A-Z]*)([A-Z])/g,'$3$2$1')")+r.toLowerCase())

This works because the only uppercase characters in the code .replace(/([A-Z])([^A-Z]*)([A-Z])/g,'$3$2$1') are A and Z, which are used to describe the character ranges. This is precisely what we need to transform into lowercase in order to process the second pass.

Test cases

let f =

s=>eval("s"+(r=".replace(/([A-Z])([^A-Z]*)([A-Z])/g,'$3$2$1')")+r.toLowerCase())

console.log(f("lLEhW OroLd!"));
console.log(f("rpGOZmaimgn uplRzse naC DEoO LdGf"));
console.log(f("eIt uqHKC RBWOO xNf ujPMO SzRE HTL EOvd yAg"));
console.log(f("NraWgCi: Nsas-eNEiTIsev rNsiTG!!"));
console.log(f("(^_^)"));
console.log(f("AWCTY HUOS RETP"));
console.log(f("hwn oeesd acsp nawyya"));
console.log(f("SpMycaeIesKyBorekn"));
console.log(f("D's mroyr, Ivam. I'e faardi I act'n od htta."));

Python 2, 181 bytes

Much longer than it should be but anyway:

def F(s):
 for l in[i for i,c in enumerate(s)if c.isupper()],[i for i,c in enumerate(s)if c.islower()]:
  for a,b in zip(l[0::2],l[1::2]):s=s[:a]+s[b]+s[a+1:b]+s[a]+s[b+1:]
 print s

This first creates two lists: one of the indices of the uppercase characters and one for the lowercase characters. Each of these lists is looped through in pairs of indices, and the characters at those indices are switched.

I'll golf this down tomorrow, but for now it's time to sleep.

PHP, 108 bytes

$f=preg_replace;echo$f("/([a-z])([^a-z]*)([a-z])/",$r="$3$2$1",
$f("/([A-Z])([^A-Z]*)([A-Z])/",$r,$argv[1]));

The code is wrapped here to fit the available space.
It can be executed from the command line:

$ php -d error_reporting=0 -r '$f=preg_replace;echo$f("/([a-z])([^a-z]*)([a-z])/",$r="$3$2$1",$f("/([A-Z])([^A-Z]*)([A-Z])/",$r,$argv[1]));' 'lLEhW OroLd!'

A 1-byte shorter version is possible on PHP 7 by squeezing the assignment of $f inside its first call:

echo($f=preg_replace)("/([a-z])([^a-z]*)([a-z])/",$r="$3$2$1",
$f("/([A-Z])([^A-Z]*)([A-Z])/",$r,$argv[1]));

Ruby, 81 bytes

puts f=->(i,s){i.gsub /([#{s})([^#{s}*)([#{s})/,'\3\2\1'}[f[$*[0],'a-z]'],'A-Z]']

Qbasic, 453 bytes

LINE INPUT a$:IF a$=""THEN END
b=len(a$):FOR a=1TO b
t$=MID$(a$,a,1):SELECT CASE t$
CASE"A"TO"Z":b$=b$+"U":u$=u$+t$
CASE"a"TO"z":b$=b$+"L":l$=l$+t$
CASE ELSE:b$=b$+"O":o$=o$+t$
END SELECT:NEXT
FOR x=1TO b STEP 2:g$=g$+MID$(u$,x+1,1)+MID$(u$,x,1):h$=h$+MID$(l$,x+1,1)+MID$(l$,x,1):NEXT
FOR x=1TO b:t$=MID$(b$,x,1)
IF t$="U"THEN
u=u+1:z$=z$+MID$(g$,u,1)
ELSEIF t$="L"THEN l=l+1:z$=z$+MID$(h$,l,1)
ELSE o=o+1:z$=z$+MID$(o$,o,1)
END IF:NEXT:?z$

This basically consists of three parts:

• Splitting the input into 4 strings (Uppercase, Lowercase, other chars and a map)
• Flipping the uppercase and lowercase letters
• Using the map to (re)construct the output.

A more detailed explanation:

' --- Part I: Reading the input
LINE INPUT a$:IF a$=""THEN END
'This FOR loop takes one character at a time
b=len(a$):FOR a=1TO b
' And checks in what category the character belongs
t$=MID$(a$,a,1):SELECT CASE t$
' For each group, char t$ is added to that group (u$ for uppercase, 
' l$ for - you've guessed it - lowercase and o$ for the rest
' And the map in b$ is updated with a U/L/O on this index.
CASE"A"TO"Z":b$=b$+"U":u$=u$+t$
CASE"a"TO"z":b$=b$+"L":l$=l$+t$
CASE ELSE:b$=b$+"O":o$=o$+t$
END SELECT:NEXT

' --- Part II: Swapping within case-groups
' Loop through u$ and l$ twp chars at a time, and add those chunks in reverse order
' to g$ and h$. Because mid$ doesn't fail past the end of a string (but returns ""), 
' this automatically compensates for odd-length groups.
FOR x=1TO b STEP 2:g$=g$+MID$(u$,x+1,1)+MID$(u$,x,1):h$=h$+MID$(l$,x+1,1)+MID$(l$,x,1):NEXT

' --- Part III: Read the map to put it all back together
FOR x=1TO b:t$=MID$(b$,x,1)
' See what group was in this spot, then read the next char from the flipped string.
' This keeps an index on those strings for the next lookup.
IF t$="U"THEN
u=u+1:z$=z$+MID$(g$,u,1)
ELSEIF t$="L"THEN l=l+1:z$=z$+MID$(h$,l,1)
ELSE o=o+1:z$=z$+MID$(o$,o,1)
' And finally,display the end result.
END IF:NEXT:?z$

ES6 155 - 195 bytes

I know there's already a better answer, but I wanted to try without regex. This one works on punctuation too, but that seems to violate the (^_^) test. In that case I have another c() function, given below.

f=(s)=>{d={};s=[...s];for(i in s){b=s[i];for(j in s)if(i<j&!d[i]&c(s[j])==c(b)){d[j]=1;s[i]=s[j];s[j]=b;break}}return s.join('')}
c=c=>~(c.charCodeAt()/32)

f("M I'o DaG") //> I'M a GoD
f("(^_^)")     //> )_^^(

c=c=>((c!=c.toUpperCase())<<1|c!=c.toLowerCase())||c.charCodeAt()

f("M I'o DaG") //> I'M a GoD
f("(^_^)")     //> (^_^)

Explanation

f=(s)=>{
    d={};        //list of indexes already swapped
    s=[...s];        //string to array, stolen from above ES6 answer
    for(i in s){
        b=s[i];        //keep a note of what we are swapping
        for(j in s)        //iterate over the array again
            if( i<j & !d[i] & c(s[j])==c(b) ){
                        //only pay attention after we pass i'th
                        //only swap if this char hasn't been swapped
                        //only swap if both chars in same 'category'
                d[j]=1;        //note that the latter char has been swapped
                s[i]=s[j];
                s[j]=b;
                break        //avoid swapping on the same 'i' twice
            }
    }
    return s.join('')        //return as string
}

Perl 6, 56 bytes

{for "A".."Z","a".."z" ->@c {s:g/(@c)(.*?)(@c)/$2$1$0/}}

Takes a string variable as an argument, and modifies it in-place so that after calling the lambda the variable contains the result.

Longer than it would be in Perl, because:

• The new regex syntax is more verbose, e.g. writing out the character classes would look like <[A..Z]> instead of [A-Z].
• Regexes are first-class source code parsed at compile time, and a string can only be interpolated into them at run-time if it consists of a self-contained subregex (i.e. you can't interpolate a string into a character class).
• Explict EVAL, which would allow more flexibility, requires the golf-unfriendly use MONKEY-SEE-NO-EVAL; pragma.

On the plus side, an array in a @ variable can be referenced directly in a regex, and is treated as an alternation.

Perl 6, 65 bytes

{reduce ->$_,@c {S:g/(@c)(.*?)(@c)/$2$1$0/},$_,"A".."Z","a".."z"}

Functional version (outputs the result as the return value of the lambda).