All antennas are designed to transmit/receive a desired polarization while the cross (or ortho) polarization is rejected. Depending upon design, the rejection can be >30 dB with a good antenna, so two pairs of antennas, each with "its own" polarization, can use one frequency for two independent "channels" with >30 dB separation.
So you should not expect "no power" to be received by the "other polarization" antenna, but a finite degree of rejection. Same principles for circular polarization.