Is there a useful mathematical way to describe the "gain" of an XOR gate on analog inputs from a large-signal perspective, on periodic signals?

Two periodic signals of some fundamental frequency and amplitude go in, and you get a pulse-density signal out of some average value. There's a multiplier effect so you get sum and differences of the fundamentals plus harmonics. Is there a way to derive the amplitudes of the individual output components.

In small signal analysis when used as e.g. a phase comparator I believe it's just assumed to have some linear gain in volt/radian increasing from 0 at a pi/2 offset between the signals, to maximum at pi and negative pi