A good way to visualize the difference between even and odd-order distortion harmonics is to imagine a sine wave ... a perfect, happy little sine wave. That's the original signal.
* Now, use a single diode to clip one side of it ... say, the positive side. In addition to generating a DC offset, if you run a spectral analysis of it, you'll see a series of harmonics ... 2nd, 4th, 6th, etc. etc. If you look at the "transfer curve" ... a curve mapping the input/ratio at different levels ... you'll see a diagonal line that is perfectly straight (the linear portion) that also has a sharp bend in it at the top, with a flat-topped region beyond the bend. The transfer curve is actually the true distortion mechanism; the spectral analysis of it is an indirect indicator that is (relatively) easy to measure.
* Let's use two diodes to clip the top and bottom sides, both positive and negative. This is known as symmetric clipping. If the flat-topping is at exactly matched levels, there will no DC offset. Similarly, if the clipping is precisely symmetric, the spectral analysis now shows no even-order terms (2nd, 4th, 6th, etc) but only odd-order (3rd, 5th, 7th, etc.). The transfer curve now has TWO kinks in it, at matching plus and minus signal levels. If the levels precisely match, there will be no even-order terms, but odd-order terms are abundant.
The diodes create hard clipping. Vacuum tubes, properly biased, create a softer "knee" region, but rest assured distortion is still there, just not as much, and with less high-order content. The property of symmetric circuits is they cancel transfer curves that are precisely opposite in shape ... a "C" shape that is inverted in the other phase of the circuit. But that depends on symmetry and precise phasing that tracks as levels go up and down.