What is the effect of antenna cross-polarization over a line-of-sight VHF or UHF path?
Why Antennas transmit and receive best when their electric field orientation matches. If one station uses a horizontal antenna and the other vertical, the polarizations are 90 degrees apart and only a small fraction of the energy couples into the receiving antenna, typically costing 20 dB or more on a direct line-of-sight path. That shows up simply as a much weaker received signal, not as distortion.
Watch out Echoes and inverted sidebands come from other mechanisms: multipath reflections cause flutter or ghosting, and sideband inversion is a mixing or modulation issue, neither of which is caused by antenna orientation. Saying nothing happens is only close to true for long skywave paths where the ionosphere scrambles polarization anyway.
Mismatched polarization = big signal loss. That's why FM repeaters are vertical and SSB/CW weak-signal work is horizontal.