As a general rule, yes! Here, this study linked in your Wikipedia article ought to do:
http://milbert.com/Files/articles/TvsT/tstxt.pdf
I think that paper has some problems, plus it was written in 1972 so some of the test methodology could be done better with modern tools. The latter obviously isn't the author's fault, but his general method was to measure distortion 12dB above the point of 1% THD. That's not particularly fair to the opamp, because opamps are designed not to distort at all until they don't have any choice in the matter, then they hard clip. (By the way, opamp is a circuit topology, not a device, although he does later specify monolithic opamp--those were fairly primitive in 1972). So an amp that is designed to distort should never use an opamp to do that, you'll always get mostly symmetrical hard clipping (hard being much worse than symmetrical).
Anyway, pay special attention to Fig. 4, 6, 8, and 10. What is the dominant distortion in fig. 6? Isn't that a tube? Why is fifth-order and seventh-order distortion more prominent in fig. 10 vs. fig. 8?
Here's some more on transformer saturation:
http://www.opamp-electronics.com/tutorials/core_saturation_2_09_11.htm
Look towards the bottom, there is a little hint on DC causing asymmetrical distortion in a transformer. That's the only way it can happen. I admit I haven't made a careful study of that behavior in guitar amp output transformers, but I don't think it would ordinarily be a large effect (due to limited DC), and I'm pretty sure that couldn't occur in push-pull designs. Some discussion on that here:
http://music-electronics-forum.com/t16762/
So a single-ended transformer output can distort asymmetrically, but we already know that a single-ended tube output is going to potentially have a large amount of even-order distortion all by itself. So do all push-pull output tube amps sound like crap? Discuss.
Again, topology: more important than device selection.
I will give you the further hint that it's not even- vs. odd- which is so compelling, but low vs. high. That's the real issue. People like third-order harmonics; ninth-order, not so much. Transistor designs tend to use lots of transistors to make more linear circuits. They have low distortion of all types--until they clip, which they will do hard and fast. You can't use 20 tubes in a tube amp due to cost and maybe heat, but if you wanted to, you could make an extremely linear tube amp (until it clipped). Actually you could do much better with even six tubes than is commonly done.
If you don't want a linear amp . . . don't do that!