I am not sure about the physics of this.

The 90deg phase shifter has to be some sort of dispersion
device (propagation time varies with frequency), and this
will operate on the fourier components of the waveform.

The fourier components present are sin(a+b) and sin(a-b),
not the complex waveform cos(a)*sin(b).

Therefore the phaseshift will apply individually to
sin(a+b) and to sin(a-b).

Can we go back to the physical device itself, and confirm
that the IF phase shifter is shifting by 90deg (as opposed to
say 270deg) and that you can select both sidebands
properly with the add/subtract?

Stewart G3YSX