Asymmetric clipping... basically, you have your waveform, which is a wiggly bunch of oscillating nonsense. If you're willing to accept oversimplification to get the basic idea down, you can view it basically as the graph of some function of time, and view clipping of any kind as a new function that behaves just like the old one for 'small' oscillations but won't allow anything past a fixed 'height' from ground. If you want to call your old waveform 'function' f, the clipped waveform will be more along the lines of min(f,c) (where c is a constant, the clipping threshold of the device) in the case of symmetric clipping. Asymmetric clipping differs from this in that the aforementioned clipping thresholds at the 'top' and 'bottom' can be different... (To view the asymmetrically-clipped waveform in the way described above, split the function f into positive and negative components - f+ = max(f,0) and f- = f - f+, so f+ is only positive, f- is only negative, and f = f+ + f- - and make the new function from the sum min(f+,c) + max(f-,c') where c and c' are the 'thresholds'.)
The attack of the note will probably be at a significantly higher volume than the decay, and you might just be hearing noise or fretting as the 'sizzle' near the end (if a pedal stops clipping at volume A, and volume A is greater than volume B, then it shouldn't clip at volume B), but to get more of that I'd just experiment with a harder pick attack. Unless something is wrong with the pedal or your ears (entirely possible, since a lot of 'dirty' things appear clean to us), that sounds like a funny thing to have happen.
The new handwired TS808s seem quite a bit like all old handwired TS808s in that they're overpriced things based on a cheap, simple circuit whose only hope for sale is the relentless belief in magic that guitarists seem to hold closer than the average mystic cult. If you want a handwired TS808, roll your own.
Jeff