I think the symmetrical/asymmetrical thing is frequently misunderstood when diodes and high gain are involved.
The use of two discrepant back-to-back diode paths (e.g., 2+1, LED+Si, etc.) means that one side/half-wave will clip under conditions of less signal amplitude than the other. IF the signal is of a sufficient amplitude that the likelihood of clipping is significantly greater for one half-cycle than the other, then there will be a noticeable difference in the distribution of harmonic content. If we start applying more and more gain, however, what eventually happens is that BOTH half-cycles smash their "heads" against the limit set by the diode, and what you end up with is a kind of symmetrical clipping, except that one half-cycle is a lower amplitude than the other by the difference between the diode clipping thresholds. So, if I have a 1N914 with a 537mv threshold on one side, and a 1N60 with a 219mv threshold on the other, then one half cycle will be 318mv greater in amplitude than the other. Still equally square, but more amplitude. What we have now is perhaps describable as the same distribution of harmonics as symmetrical clipping, but a slightly different weighting.
The key thing here is that "asymmetry" is a function of signal amplitude, and "real world" guitar signals vary so widely in amplitude as to present very different outcomes of signal-vs-diodes on a moment-to-moment basis.
Another aspect of discrepant diode paths is that it almost invariably results in greater *potential* for signal amplitude by raising the threshold for one half cycle. So, tack on another 1N914 to a TS-9 to make it work like an SD-1, and you have not just made it asymmetrical but have raised the clipping threshold for one half cycle so that increases in picking strength are still audible. People talk about greater dynamic responsiveness of the SD-1 vs the TS-9, or complained about the "compressed sound" of the TS-9. However, the greater dynamic responsiveness would still be evident if you used a 2+2 diode complement in a TS-9, rather than the 2+1 of an SD-1. In other words, what they attribute to asymmetrical clipping is is actually a product of having a higher clipping threshold for one half-cycle. You will note that it is very rare that people adapt devices for asymmetrical clipping by switching from, say, a pair of 1N4148's to a 1N34 and (even lower threshold) Schottky diode. They always "go asymmetrical" by increasing the threshold on one side of the diode path. What they like is the tendency to not generate as much harmonic content until they really "lean into" a note, but that is all about higher threshold, not the asymmetry of the threshold.
Again, I'm not dismissing distortion symmetry as a myth or foolishness. Rather, the way in which it is usually implemented misleads many into thinking that what they like about the change is DUE to asymmetry and not to something changed by the addition of diodes, and consequent change in clipping threshold.