Thanks, Jon - great explanation. 
So if "Push-pull amplifiers (tube and solid state) are able to cancel this asymmetric distortion, and thus tend to contribute symmetric distortion (3rd harmonic) - although in much lower levels", does that mean that push-pull tube amps produce 3rd harmonic distrortion just like ss amps do? In fact they may well produce higher levels?
And what about 5th / 7th HD etc?
Thanks,
Andy
Happy to help

Push-pull amps generally produce third harmonic as dominant *if* they don't include any single-ended gain stages, and *if* they don't have imbalance between the signals in each of the push-pull "halves", and *if* the output transformer is not saturating or experiencing DC offset, along with a few more qualifications. Push-pull signal imbalance (one of the biggest design concerns) contributes to increased 2nd harmonic due to - you probably can guess it - increased asymmetry. Many push-pull tube amp circuits also include a single ended triode or pentode input stage, which can also contribute some 2nd HD. Higher harmonics can be produced, too, depending on the tubes being used and how they are implemented, but generally at lower magnitudes - and things do get weirder when various types of feedback are used. You should really expect to see a mix in almost any tube amplifier, with 2nd and 3rd dominating in most cases for SE and push-pull amps, respectively.
BTW, some of the single-ended amp assumptions I make do not necessarily apply to pentodes, where it is possible to specify an operating point where the 2nd harmonic nearly vanishes - With careful tuning, you can make a pentode amp's distortion signature look like almost anything you want (although I'd usually prefer it to be as weak and low-order as possible

).
Also, comparing push-pull solid state to push-pull tubes - the magnitudes really can be quite comparable, and the tubes often produce "cleaner" (lower-order) distortion components. My buddy and I are getting about 0.02% THD, mostly 3rd order and some noise, at 1 kHz / 1 Watt for our latest amp projects. It may be a bit lower in the final version after some tuning, targeting the 0.005-0.015% range.