We report the experimental observation of oscillatory antiphase switching between counterpropagating light beams in Kerr ring microresonators, including the emergence of periodic behaviour from a chaotic regime. Self-switching occurs in balanced regimes of operation and is well captured by a simple coupled dynamical system featuring only the self-and cross-phase Kerr nonlinearities. Switching phenomena are due to temporal instabilities of symmetry-broken states combined with attractor merging that restores the broken symmetry on average. Self-switching of counterpropagating light is robust for realising controllable, all-optical generation of waveforms, signal encoding and chaotic cryptography.