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Re tube amps, it depends on the design and topology. I gather that the output stage of a well designed push-pull actually cancels even harmonics, leaving predominantly odd, so if you want them you need to add them in the pre and driver circuits.
Pentodes and beam-power tetrodes produce plenty of odd harmonics, as well as even. Like dougi said, push-pull cancels even order to some extent, depending on the matching of the tubes and the balance of the transformer. It's a complicated subject, with no simple answer.
Amplifier design has far more impact on distortion characteristics than the gain devices used. It's possible to design low distortion tube amplifiers and high distortion solid state amps.
Tube power amps usually need output transformers, and these have challenging design characteristics, especially at the frequency extremes and as they approach saturation.