Hi Denis, yes sure, here is a short description of the game.
In the game, for which, together with two programmers from Germany we made an implementation for on the iPhone, 2 players put in turn black and white stones on the board. The symmetry element comes in when a move by a player causes a square on the board (which could be of any size 2x2, 3x3 etc) to be fully covered by stones. If because of this move the color pattern of this square is also ´mirror symmetric´, horizontally or vertically (I exclude ´diagonal´ symmetries), then all stones in that square change into the color of the player who just put the stone on the board. And of course, in the end, when the board is full with stones, the player with the most stones of her/his color wins.
So In this way the game quite directly uses the concept of symmetry (We also named the game "Symmetry - The Board Game"
) . Now this is a game, but I could imagine that this ´reasoning whith symmetries' would be something that lends itself very well for puzzles, but as far as I know I´ve never come across such puzzle.