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.

If both antennas simultaneously received a direct signal from the same source, the two IQ pairs (four numbers) would would include redundant information, since one could define three numbers, phi, r1, and r2 such that

i1 = cos(phi) * r1
q1 = sin(phi) * r1
i1 = cos(ph2) * r2
q1 = sin(ph2) * r2

In practice, however, there's no guarantee that signals will only arrive via a direct path. It's possible that an antenna may receive reflected signals as well. Imagine that the transmitter and receiver are placed so that a direct signal is blocked, but reflective surfaces exist on either side; a signal reflected off one side will arrive at the receiver from a direction perpendicular to that of a signal reflected off the other. Depending upon the exact distances, the signals reaching the antennas could have any possible phase relationship. While it might be advantageous to determine a couple of parameters phi1 and phi2 and then compute a master output signal as:

net = (I1*cos(phi1)+Q1*cos(phi1))*cos(phi2)+(I2*cos(phi1)+Q2*cos(phi1))*sin(phi2)

Determining what the parameters phi1 and phi2 should be is apt to be difficult unless one has all four IQ signals to work with.

Generally speaking, the voltage at the output of an antenna is just a scalar function of time, so you of course don't need anything more. You can connect each antenna input to a suitably fast wideband digitizer, and you'll be able to receive the signals directly. But such digitizers, of sufficient performance, may either not exist, or cost way too much, or there may not be processing power available to process such a datastream in real time, etc.

So, instead, you mix the signals to shift them to either a lower intermediate frequency or to the baseband, and process the result of such shifting a bandwidth much narrower than the frequency of the incoming signal. You'd typically design the signal path so that the passband is centered around the nominal output frequency (IF or 0Hz for baseband). When you do that, you won't be able to distinguish positive from negative frequencies with just a scalar signal, relative to the IF frequency, and in the case of baseband (or sub-Nyquist sampling) you'll also lose the signal as the relative phase of the signal approaches 0 degrees. So instead, you'd need to filter the passband so that it doesn't overlap the "center" frequency, i.e. 0Hz or the IF, and digitize that instead. Such circuits usually cost more, since you still have the symmetric passband capability - a circuit with 3MHz bandwidth doesn't care if it processes IF+1.5MHz or IF-1.5MHz; but you'll use less than half of the bandwidth available; conversely if you have a circuit that can in general process IF+1.5MHz, it will is fast enough to process IF-1.5MHz anyway, so you may as well use that.

So yes, if you wished to you certainly can design such a system, and in low quantities the cost difference probably wouldn't matter much, although things may get out of hand quickly as you approach the state of the art. It will usually also be the case that the cost and effort needed to design the circuit that has twice the bandwidth will be more than the cost of having two quadrature channels at half the bandwidth, especially if the mixer just provides you with I and Q anyway.

You also seem to be very concerned about the number of signals, whereas the overall design driver would be overall bandwidth in this case - you'll end up pushing the same amount of data to your processor no matter how you slice and dice those signals, it's just that the less signals you have the more demanding each one will be by itself (in terms of bandwidth both analog and digital). None of it may be a big concern if your overall bandwidth is "low" (say gigabit or less after digitization) - any reasonable PC will keep up with that if you do it right.