Make this dice game fair
Jelly, 34 bytes

+þµFṢ
ç©ṀRœċL}Œċµç/⁼®µÐf
,Ṣ€,Ṛ$ḟ@ç


Returns a list of all possible pairs of dice up to ordering of dice and faces* (excluding any that are the same dice as the input pair).

Try it online!

This is a brute-forcer, and is not efficient (too slow to even run the 6-face test case at TIO).

+þµFṢ - Link 1, sorted sums: dieA, dieB
 þ    - outer product with
+     -     addition (make a table of the sums, a list of lists)
  µ   - monadic chain separation
   F  - flatten into one list
    Ṣ - sort

ç©ṀRœċL}Œċµç/⁼®µÐf - Link 2, all possible fair dice pairs (including input): dieA, dieB
ç                  - call the last link (1) as a dyad
 ©                 - save the result to the register for later use too
  Ṁ                - maximum (the largest sum)
   R               - range [1,2,...,maxSum]
       }           - use right argument (dice B) to turn the monad into a dyad:
      L            -     length
    œċ             - all combinations with replacement (i.e. all dice with faces between 1 and maxSum inclusive [overkill, but golfier than less])
        Œċ         - all pairs of these dice
          µ        - monadic chain separation
               µÐf - filter keep:
            /      -     reduce with:
           ç       -         last link (1) as a dyad (the sorted sums of the current pair)
              ®    -     retrieve register value (the sorted sums of the input pair)
             ⁼     -     equal?

,Ṣ€,Ṛ$ḟ@ç - Main link: dieA, dieB
,         - pair - [dieA, dieB]
 Ṣ€       - sort €ach - [dieASorted, dieBSorted]
     $    - last two links as a monad:
    Ṛ     -     reverse - [dieBSorted, dieASorted]
   ,      -     pair - [[dieASorted, dieBSorted], [dieBSorted, dieASorted]]
        ç - call the last link (2) as a dyad (with dieA, dieB)
       @  - reversed @rguments:
      ḟ   -     filter out - removes any results that are equivalent to the input pair.


* i.e. if [[1,1,4,4],[1,1,5,5]] is a possibility (like in one of the examples) there wont be a [[1,1,5,5],[1,1,4,4] or [[1,4,1,4],[1,1,5,5]], etc.