Ok, you didn’t mention about horizontal polarized in your first post. Cross-polar and Horizontal-polar are different.
A horizontal polarized antenna would be exactly like a vertical polarized antenna, but spinned by 90°. Like a V-pol antenna laying on its side… It can be used, but the consequences of such a setup are obviously not useful in most situations :
…. horizontal lobe would only be 7° to 14°, while the vertical lobe would be about 65° to 90°. As you understood, the lobes are inverted, compared to a vertical polarized antenna. Not very practical. Square antennas have about the same H-lobe than V-lobe (30° or 65°… depends).
I wonder if you’re actually talking about space diversity, achived by horizontal separation or vertical separation. Let me know…