In this paper, we propose and demonstrate time-modulated patch antennas able to exhibit opposite polarization ellipticity when operated in transmission or reception, effectively leading to nonreciprocal polarization responses. To this purpose, we merge a patch antenna fed from four symmetrical sides with a low-frequency time-modulation scheme. This configuration exploits the photonic Aharonov Bohm effect to individually manipulate the phase of surface currents flowing along orthogonal directions on the antenna with the phase of the modulation signals. Experimental results at 2.2 GHz demonstrate high conversion efficiency in the time-modulation process (loss 3 dB), isolation levels over 40 dB in transmission/reception mode, and tunability to generate/receive electromagnetic waves with arbitrary polarization ellipticity. Our findings may enable exciting applications in full-duplex communications as well as in polarimetric radar, sensing and imaging systems.