We introduce an optical system (a coupler) obeying parity-time (PT) symmetry with odd-time reversal, T^{2}=-1. It is implemented with two birefringent waveguides embedded in an anti-PT-symmetric medium. The system possesses properties that are untypical for most physical systems with the conventional even-time reversal. Having a symmetry-protected degeneracy of the linear modes, the coupler allows for the realization of a coherent switch operating with a superposition of binary states that are distinguished by their polarizations. When a Kerr nonlinearity is taken into account, each linear state, being doubly degenerated, bifurcates into several distinct nonlinear modes, some of which are dynamically stable. The nonlinear modes are characterized by amplitude and by polarization and come in PT-conjugate pairs.