Can I/Q signals from two perpendicular antennas be simplified to any less than 4 signals?
You need all four signals.

I/Q processing is there to keep you from getting shafted by phase cancellation between your received signal and your local oscillator (LO) when you heterodyne to baseband.

Consider: Assume your LO and your received signal are on the same frequency, but have a phase offset.  Mixing the two results in an output signal, with and attenuation from phase cancellation that is proportional to \$(1 + \sin \omega)\$, where \$\omega\$ is the phase offset angle in radians.  If the two signals are 180 degrees out, they cancel completely.

I/Q processing uses two mixers and an LO that emits two signals, on the same frequency but 90 degrees out-of-phase.  If the received signal is 180 degrees out from one LO signal, it will only be 90 degrees out from the other, and that side of the I/Q pair will still have output.

As a freebie, you get phase information.  The degree of cancellation on the two signals tells you the exact phase angle of the incoming signal against your LO.

This is EXACTLY the same principle as your two loop antennas, rotated 90 degrees.  If the signal you are interested in is on the null from one antenna, you still get signal from the other antenna, and now you know where the signal is coming from.

That's why you do full I/Q processing on both antenna signals, which means you need all four signals.