Keep in mind that an amplifier does not need to be single ended (one tube) to be considered "class A" and that
class A push/pull amplifiers exist. The main definition of "class A" is that the output stage is biased so that the output devices conduct through all 360 degrees of the wave. In a class A push pull, one side is designed to favor the positive side, and the other the negative side, but both sides conduct the entire wave form at all times. What this means is that "class A" is not a design so much as it's a
mode of operation. In other words, a single tube amp MUST be class A, but one or two or more evenly matched pairs of tubes can be operated in class A mode.
In a class B amp, the tubes do exactly 180 degrees of the signal. In an AB amp, the designer sets it so that the tubes conduct a bit beyond 180 degrees, say 190, and this makes it get some of the characteristics of the class A operation, and some of the B in one amp.
The reason, besides efficiency, that class AB was created was so that the amp designer could the minimize crossover distortion of class B, as well as retaining distortion characteristics...while also lowering them...of the class A design (even order harmonics). Most all of the amplifiers designed back then were made to minimize distortion of any kind.
It took Jim Marshall, and Randall Smith to finally make the amp designer world embrace distortion as an end unto itself.
Here's a description that's better than I could come up with.
http://www.audioholics.com/audio-amplifier/amplifier-classes
"As compared with the other amplifier classes we’ll cover, Class A amps are relatively simple devices. The defining principle of Class A operation is that all of an amplifier’s output devices must be conducting through the full 360 degree cycle of a waveform. Class A can also be broken down into single ended and push/pull amplifiers. Push/pull diverges from the basic explanation above by utilizing output devices in pairs. While both devices are conducting through the full 360 degree cycle, one device will shoulder more of the load during the positive portion of the cycle, while the other handles more of the negative cycle; the primary advantage of this arrangement is reduced distortion relative to single ended designs, as even order harmonics are cancelled out. In addition, push/pull Class A designs are less susceptible to hum; single ended designs tend to require special attention to the power supply to mitigate this issue."