Look at this experiment:
symmetric vs asymmetric clipping.pd

The osc~ frequency is set so that it perfectly fits a 64 sample block, which make the FFT’s terms match the harmonics of the input.

When I increase the top and bottom clipping amount, the wave shape approaches a square wave and the harmonic content matches theory—all odd numbered harmonics, meaning harmonics that are odd multiples of the fundamental. Now I’ve read that even harmonics are emphasized in asymmetric signals, harmonics that are supposedly warmer and richer and are of the kind produced by single-ended tube stages, all of which have asymmetric current gain curves. That’s what my “top only” slider is crudely simulating. But the FFT contains both even and odd harmonics and there doesn’t seem to be any bias towards even harmonics, unless their mere presence is the magic sauce. What am I misunderstanding?

Edit: I was curious whether my heavy-handed asymmetric clipping was at fault, but this gentler version has similar (if not worse) characteristics.
symmetric vs asymmetric clipping 2.pd