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JavaScript (Node.js),  110 ... 97  96 bytes

c=>(s=")0@2^6&7*8(9\"'+=-_:;`~")[s.indexOf(c)^1]||(B=Buffer)([([n]=B(c),n%127>32)<<4+n/64^n])+''

Try it online!

How?

We use a lookup table for these pairs:

) ↔️ 0
@ ↔️ 2
^ ↔️ 6
& ↔️ 7
* ↔️ 8
( ↔️ 9
" ↔️ '
+ ↔️ =
- ↔️ _
: ↔️ ;
` ↔️ ~

For all other characters, we use the following code:

(B = Buffer)([   // generate a buffer from a singleton array:
  (              //
    [n] = B(c),  //   n = ASCII code of the input character
    n % 127 > 32 //   1 if 32 < n < 127, or 0 otherwise
  )              //
  << 4 + n / 64  //   left-shifted by 4 if n < 64, or by 5 otherwise
  ^ n            //   XOR'ed with n
]) + ''          // end of Buffer(); coerce it to a string

which supports these 3 cases:

• If \$32<n<64\$, we XOR the ASCII code with \$16\$, which is what we need for:

  ! ↔️ 1
  # ↔️ 3
  $ ↔️ 4
  % ↔️ 5
  , ↔️ <
  . ↔️ >
  / ↔️ ?
  

• If \$64\le n<127\$, we XOR the ASCII code with \$32\$, which toggles the case of letters and also works for these pairs:

  [ ↔️ {
  \ ↔️ |
  ] ↔️ }
  

• If \$n\le32\$ or \$n=127\$, the character is left unchanged (ASCII code XOR'ed with \$0\$).

x86-16, Genuine IBM PC, 32 31 29 bytes

00000000: b4f0 8ec0 bfe5 e8b3 3ab1 74a0 8200 f2ae  ........:.t.....
00000010: 3acb 7f02 f7db 268a 41ff cd29 c3         :.....&.A..).

Build with xxd -r.

Unassembled listing:

B4 F0       MOV  AH, 0F0H               ; BIOS segment address (F000H)
8E C0       MOV  ES, AX                 ; set ES to BIOS segment for SCASB
BF E8E5     MOV  DI, E8E5H              ; set to LABEL K10 
B3 3A       MOV  BL, 58                 ; size of table 
B1 74       MOV  CL, 58*2               ; size of both tables 
A0 0082     MOV  AL, BYTE PTR DS:[82H]  ; get input char from command line 
F2/ AE      REPNZ SCASB                 ; search BIOS table for char 
3A CB       CMP  CL, BL                 ; found in the lowercase table? 
7F 02       JG   UPPER                  ; if so, convert with uppercase table by adding
F7 DB       NEG  BX                     ; otherwise subtract the offset 
        UPPER: 
26:8A 41 FF MOV  AL, ES:[BX+DI-1]       ; get char from table
CD 29       INT  29H                    ; DOS fast write to console 
C3          RET                         ; return to DOS

Input/Output:

How does it work?

Well, the PC BIOS already contains all of the code and tables necessary for this since it actually handles the conversion of scan codes received from the keyboard to ASCII chars (in real/DOS mode at least). There's no way (that I know of) to actually hook this on just any "PC-compatible" BIOS, however if you know the location of the table in ROM (on the IBM PC it starts at F000:E8E5), you can use that.

On Page A-25 of Appendix A the IBM PC Technical Reference (listing of the entire source code of the PC BIOS) is the disassembly of this table:

Unfortunately, this address will be different on any given PC-clone BIOS and of course, there's no guarantee that it would even be implemented in the same way by anyone else. Thus, this submission is guaranteed to run only on a Genuine IBM PC. Go Big Blue!

Retina 0.8.2, 34 bytes

T`-~A-]d';-?/.+,:"(*&^%$#@!)}-_`Ro

Try it online! Link includes test cases.

Explanation: The T command is Retina's transliteration operation. Among its special characters are:

• `: Delimit the parts of the command
• -: Introduce a range (this suppresses the following special characters so d-o just means defghijklmno)
• _: Delete the matching character
• d: 0-9
• R: Reverse the next range
• o: Take the source string as the range

The trick is therefore to create as many ranges as possible to reduce the length, plus avoiding having the - and _ characters where they would be interpreted as a special character and avoiding the ` at all by including it inside a range. Additionally arranging for the range to transliterate to its reversal allows that to be expressed very cheaply.