Actually, I'm saying the multipath you describe (time-invariant coherent interference) should not produce what you saw. If I understand the physics correctly...
If you consider a signal transmitted at a single frequency, you can scatter it however many times from however many static (not changing in any way with time) scatterers, but when you recombine all those linearly at any point (specifically the receiver), the sum will have the same frequency as the signal originally transmitted. Only the amplitude and phase will be different in the final sum signal. I think you can effectively consider any of the low-symbol-rate modes single-frequency for this purpose. So I don't think time-invariant multipath explains what is happening here.
If your scatterers start changing slowly, you will see the sum signal at the receiver slowly changing in phase and amplitude, which would be typical fading. The frequency remains essentially constant. Again, not what you saw.
If the scatterers change rapidly, you could modulate the sum signal fast enough to create obvious sidebands. It seems unlikely there would be anything like a natural modulator generating carrier plus one sideband, which is similar to what we've seen, but I don't know the math behind that.
I suspect there's some sort of moving atmospheric refractor creating a Doppler-shifted version of the original signal, but what is it and what is its path? Maybe some sort of moving density boundary? Maybe a big blob of something that is expanding or contracting? Waves of some sort in the troposphere?Beats me, but I imagine someone with a good understanding of atmospheric science would have some ideas.
Do you remember, in the case where you saw four signals, were they equally spaced in frequency?