Is there any card game where the cards' back color is meaningful?
Nertz requires a different color back (or back design) for each player to determine the score at the end of the game, so two-player Nertz would match your description.


  Each player (team) scores one point for each of their own cards that they managed to play into the common area. To determine this, the foundation piles have to be sorted out according to the owners of the cards - this is why it is necessary that the decks have different backs.