We study a class of two-player normal-form games with cyclical payoff structures. A game is called circulant if both players’ payoff matrices fulfill a rotational symmetry condition. The class of circulant games contains well-known examples such as Matching Pennies, Rock-Paper-Scissors, as well as subclasses of coordination and common interest games. The best response correspondences in circulant games induce a partition on each player’s set of pure strategies into equivalence classes. In any Nash Equilibrium, all strategies within one class are either played with strictly positive or with zero probability. We further show that, strikingly, a single parameter fully determines the exact number and the structure of all Nash equilibria (pure and mixed) in these games. The parameter itself only depends on the position of the largest payoff in the first row of one of the player’s payoff matrix.