For the 3'rd order just shift bias of the same FET. Or diode, does not matter.
If symmetrical distortions cancel 2'nd order harmonic, it does not mean they and only they generate the 3'rd order harmonic. Squeeze a sinusoidal signal on one side and you will get the whole spectrum dependidng on order of squeezing function. Squeezing function of a diode is exponential, it means that the more you squeeze, the more power of more higher harmonics you get (example - emitter follower).
If you add harmonics with a level that does not depend on the power of signal it will sound mechanically, unnatural. In the nature the louder is the sound, the richer is it with harmonics. No exceptions.