How Exactly Does a Push-pull Pair Cancel Even Harmonics?
If you look at a push pull amplifier with some amount of cross-over distortion it looks like this: -



In other words it creates a symmetrical shaped signal in amplitude but generates odd harmonics that can be significantly reduced by correct biasing and sometimes negative feedback.

Now take a look at an example of even harmonic distortion: -



You should be able to see that it can generate significant amplitude asymmetry and this, in even a slightly badly designed push pull amplifier, is unlikely to happen to any great extent. 

It can happen (of course) if the load is more easily driven by the pushing transistor compared to the pulling transistor - asymmetry in amplitude will result but it would be a really crappy push-pull.

Compare this to a single transistor driver circuit and you will definitely witness some amplitude asymmetry: -