I've never tried this before, but I've always wanted to... for a phase inverter, you can specially order a "matched" 12AX7 (yes, the single tube can be matched within itself - triodes) from some of the specialty tube stores online. I almost bought a matched NOS mullard for my Twin, but for some reason never got around to it.
You're correct. You need a balanced tube for the phase inverter for a push pull amp. One side of the tube does the amplification above zero and the other side below zero with half the output tubes working either way. this is the same reason you need a matched set for the output tubes. Otherwise when you look at the waveform in your editor you will see a difference between the part of the wave above the line and below.