The idea is that the difficulty of the game of chess is derived primarily from the asymmetry between the king and queen. all other chess pieces are arranged symmetrically and can move symmetrically, then, if there were two queens (or two kings) instead of a king and queen for each player, the game would be exactly symmetrical. The idea is that, in this case, if the second player repeats symmetrically the moves of the first, the outcome should be a draw. But I'm not able to prove this (and perhaps it is not true) but it seems that the symmetry should introduce a simplification in the game who may moreover allow to define a strategy.
My knowledge of game theory is very poor, but I'd like to know if there are any studies on this symmetrical variant of the chess game.
If a rook takes the opposing rook, then the opponent can't keep the symmetry going.
As Winther says, if White checks Black, then Black usually can't do the symmetric move, they have to get out of check. Sometimes the symmetric move will achieve that.
There are usually dozens of possible moves at any time. So after ten moves each, there have been more than $12^{20}$ possible games, and more than a trillion with the symmetry.