Are Diode Ring Mixers Fundamentally Different?

Joop, PE1CQP, and I have been discussing mixer circuits, especially the ever-popular diode ring.
Here is my latest e-mail to Joop. The RSGB diagram for the ring diode mixer appears above.

Joop: I think the way the diode ring mixer works is very different from the way a two diode singly balanced mixer functions. The effect, of course, is the same. But the polarity reversing element introduced by the ring configuration -- it seems to me -- makes this a very different circuit.

Attached is the RSGB Handbook diagram I mentioned. I like it, because you can really SEE how the actions of the diode ring produce the sum and difference freqs (you have to keep Fourier in mind, and imagine the results of filtering).

The two diode circuit simply "chops" the input signal at the rate of the LO. And it would even work in a non-switching mode -- you could, for example, use FETs instead of the diodes and bias them to operate in the non-linear portion of their curves, right? This makes me think that the diode ring mixer circuits (aka "polarity switching mixers" or "commutating mixers") are very different.

73 Bill