A one-dimensional magnetophotonic crystal with a nonlinear defect placed either symmetrically or asymmetrically inside the structure is considered. Simultaneous effects of time-reversal nonreciprocity and nonlinear spatial asymmetry in the structure are studied. Bistable response is demonstrated in a such system, accompanied by abrupt polarization switching between two circular or elliptical polarizations for transmitted and reflected waves. The effect is explained in terms of field localization at defect-mode spectral resonances and can be used in the design of thin-film optical isolators and polarization transformation devices.