@agedhorse : I was using terms incorrectly.
The standard explanation of clipping is that it generates
harmonic distortion, odd vs. even depending on the symmetry. Similarly, the standard explanation is that more complex nonlinearities, like switching transients in class D amplifiers, generate intermodulation distortion.
Unfortunately, "clipping generates harmonic distortion, not intermodulation distortion" is only true
for a single frequency. Once you add multiple frequencies to the signal, you can lose enough information via clipping to generate intermodulation distortion.
This is the part I was missing. (Which I should know from emulating fuzz and distortion circuits as well as playing through them! It's why "power chords" exist: with heavy clipping, thirds can generate all sorts of ugly intermodulation. This is also the part where I facepalm.)
To return to my original point, which I hope I can make clear:
all amplifier designs can be forced into clipping. However, when @Old Fart said "I have heard that Class D amps have a certain amount of unavoidable distortion," I was reasonably sure he's talking about the type unique to Class D and its derivatives: the intermodulation distortion inherent in modulating an audio signal with the high-frequency carrier wave necessary to produce the PWM, and the switching transients inherent in real-world non-ideal semiconductors.
Is this unavoidable? Yes. Can it be minimized into practical inaudibility? Also yes,
as agedhorse pointed out.
Do class AB designs also have "unavoidable distortion" that good design can minimize into practical inaudibility? Yes, which was the point I was trying to make.
And yes, lack of crossover distortion is the primary reason for class A amplifier topologies. (Whether that's relevant or audible under real-world conditions, given a well-designed class AB amplifier, isn't a proposition I'm going to defend.) 
