A finite impartial game is a two-player game in which the players take turns making moves and the game ends after finitely many moves. In this paper, we study a class of finite impartial games introduced by H.~Lenstra, which we call coin turning games. We focus on two typical classes of coin turning games, namely the order ideal games and the rulers, distinguished by their choices of turning sets. For several posets arising from enumerative combinatorics, we determine the Sprague-Grundy functions. In particular, we determine the Sprague-Grundy function of the order ideal game on the ASM poset, introduced by J.~Striker in connection with the alternating sign matrices.