The whole thing about amp classes is kind of misunderstood.
Class A: The power device runs at full power all the time. It is more linear because the power device does not reduce to a lower current mode near the DC bias point (this is where low-level signals run, as well as the middle of high-level signal, known as crossover).
Class AB: The power device runs at low-ish power near the DC bias point, and cranks up when needed to reproduce big signals. It is WAY more efficient than Class A, so it runs at a cooler temp and is less expensive to make and operate. Almost ALL amplifiers (even Hi-Fi amps) are Class AB for these reasons.
The twist: you can adjust the design of a class AB circuit so that it runs anywhere from NO current at the crossover (class B) to full current at the crossover, at which point it's exactly the same as class A.
The less current you run at bias/crossover, the more CROSSOVER DISTORTION you get. This is the result of running low current in the power devices, which causes them to lose gain. You can also get mismatches in push-pull devices that cause additional offset operation, like dead-bands, at the crossover. This is why matched tubes are a good idea.
In a well-designed amp, crossover distortion is very small, and probably not very audible.
In Hi-Fi amps, the power section is in a FEEDBACK LOOP, so the gain of the amp reduces the artifacts, like distortion. This is why class AB is OK in Hi-Fi.
In guitar amps, the output stage is open-loop (no feedback), so the artifacts are not reduced. However, they are usually small, and probably not audible due to all the other saturation artifacts, as well as the general characteristic of a guitar signal (not a very hi-fi item). The mismatches in the output stage design and tubes can be optimized with BIAS CONTROL. This is why there are raging arguments about fixed bias vs. variable bias.
(My comment about variable bias is that tubes change rapidly over time. If you are not rebiasing the amp every month, or even more often, then you may as well have a fixed bias. The one thing variable bias gets you is the ability to run ANY tube at about the same gain as any other tube.)
Back to classes: Mesa amps probably run close enough to class A that it would be hard to tell the difference. However, they are class AB. And even in heavy class AB, it would be hard to tell the difference between AB and A with a guitar. You would need to plug in a HiFi source. AB has a bad name because a lot of amps were very poorly designed, and even well designed class AB amps sound bad with bad or badly mismatched tubes.