This study investigates piecewise-constant oscillators coupled by hysteresis elements. Coexistence phenomena of inphase and anti-phase synchronizations are observed in van der Pol oscillators coupled by inductors. Wave propagation phenomena called phase-inversion waves are observed in oscillators coupled as a ladder, a ring and so on. We focus on piecewise-constant oscillators which can be strictly and easily analyzed. In this report, we confirm by using analysis with rigorous solutions, in-and-anti-phase synchronization, in-phase synchronization and phase-inversion waves are coexisting in combination of initial values, and coexistence phenomena disappears by changing parameter corresponding to threshold value of hysteresis elements of coupled terms.