A second-order ordinary differential equation with a three-position hysteresis relay characteristic and a periodic perturbation function is considered. The existence theorem is proved for an oscillatory solution with a complete traversal of the characteristic with a possible exit into its saturation zones in some finite time and with a closed phase trajectory of an arbitrary shape. Sufficient conditions for the existence of periodic solutions with arbitrary and symmetric phase trajectories are established, as well as conditions for the nonexistence of a periodic solution with a symmetric phase trajectory. Numerical examples are given.