I was told once that a phase inverter tube should "just work" and that it has no effect whatsoever on the tone of the amplifier. Is this true?
The phase inverter is the final amplifier of the chain. If you overdive it, it will distort. The distortion depends on the tube(some tubes will distort sooner than others or differently).
A phase inverter is simply an amplifier configured in such a way to give the original amplified signal and another signal 180 degree's out of phase to drive the push pull amp.
There are two things that can make the amp sound different due to the PI. If it is overdriven and/or if it is asymmetric.
It depends a lot on the tube itself and on the type of phase inverter circuit used.
In any case the tube will have about the same effect as any other tube in the chain. Any tube will have more of an effect the harder it is driven. All tubes will behave approximately the same if they are kept in in their linear range. Even bad tubes can sound good if not driven hard enough(All physical devices behave linearly within some range).
There are more important things in a tube amp that effect the sound than the tubes(if the tubes are working). EQ, pre-amp bias type/level, Full load current, tone stack type, feedback amount, etc...
What makes tube amps sound different is generally not the tubes since all tubes pretty much behave very similarly even when distorting.
Whatever minute differences you'll get from changing good tubes don't compare to changing the EQ(EQ has the most drastic effect on an amp's distortion).
Of course we are assuming working tubes.
The reason why some tubes do work better typically have to do with the type of circuit the amp is built around rather than the tube itself. I know this sounds contradictory but the point is that some circuits happen to be out of spec and some tubes can handle it better(or worse depending on your perspective). What this means is that it's more of a problem with the circuit itself than something special about the tube and instead of changing the tubes you can fix the circuit. Of course it's easier for people to change tubes rather which is why you get so many people doing it and claiming some tubes are better than others.
Given that there are only like 4 vacuum tube manufacturers left in the world and 99% of the tubes you see are just rebranded versions goes to show that tubes don't make all that much of a difference.
The more distortion you are after though the more the tubes come into effect up to a point then all tubes will sound almost identical.
The idea is that the pre-amp is where you get your distortion by overdriving the tubes there. Unfortunately this can happen in the PI too trying to get more gain which causes them to distort. Because of the circuit difference it will distort differently which will have a different effect.
If you are comparing PI tubes make sure you are driving them as hard as the amp allows to get the real difference else they will all sound almost identical(i.e., the whole point of saying they are operating in the linear region).
Apologies for answering my own question, but after performing some empirical experiments the answer in short is: yes. A phase inverter has an effect on the tone of your amplifier. I performed some experiments with the following tubes:
I matched all the following combinations for the PA and the PI in my Orange AD-30R and recorded a clip of me playing (almost) the same riff with my Les Paul R8. The settings and volume were identical on each clip, and they were recorded through a SM-57 into an Apogee Duet with Logic Pro on my Macbook.
After I have A/B'd these clips, I can tell that there are obvious differences between all of them. Take the two clips with the JJ's for the Preamp tube for example. There is a very noticable harshness to the clip using another JJ in the PI, while the Mullard is less so--fatter and more evenly spaced. There is, of course, also the fact that the Mullard is a 12AT7 and the JJ is a 12AX7. That could attribute to the fatter sound due to higher headroom and a much lower gain rating.
As you can tell in each of these clips, there is a noticeable difference between the JJ phase inverter and the Mullard phase inverter. I have ran additional tests with the RCA and Tesla tubes as phase inverters, and in each case the amplifier sounds different—so the difference in tube types isn’t completely the reason for the tone difference.
It's much too long to quote, so I'll link to Myles Rose's paper about the Phase Inverter and quote this:
It makes sense that the phase inverter is very important. It's feeding the signal from the pre-amp into the power-tubes and has to make that signal change very cleanly, without adding any distortion. At least that's the idea for an amp running very cleanly.
With a lot of amps we rely on the pre-amp stage to provide the distortion, so the power-amp tubes really should only amplify. But, there's a really nice distortion you get when the power-tubes are getting pushed, and, according to that article it's the phase-inverter that is responsible for that sound. So, making sure that tube is in good shape is important otherwise the output-stage distortion will be weak or flabby sounding instead of fat, like it should sound.
All the tubes degenerate as they are used so it's a good idea to swap them out periodically. A good amp tech. will be your best asset when trying to figure out when that should be.