The setup and rules of orthodox chess are formed, but with the following modification.
Each turn, a player first makes a normal move with one of his pieces (or castles), and then makes a number of moves with pieces of the opponent. The number of moves increases: first white makes one move with white pieces and one move with black pieces; then black makes one move with black pieces and two moves with white pieces; then white makes one move with white pieces and three moves with black pieces, etc. So, the ith turn, the player makes one move with his own pieces, and i moves with the pieces of the opponent.
Every move made should be legal: one may not move the pieces of the opponent such that a player is in check (or checkmated) after the move.