Absolute phase, if music was all tone fundamentals, would not be detectable if switched.
But Nelson Pass has made some interesting studies where the phase of harmonics switches if you invert absolute phase. It's easy to show in a graph, but the principle is that if you have a second order harmonic in which a positive "peak" of music coincides with the positive "peak" of the second harmonic, if you revert the absolute phase, that positive peak (which now is a negative one) would still coincide with a positive second harmonic.
I'm on my iPad now, I could provide a nice graph that explain this perfectly tomorrow, but essentially, if you invert absolute phase, you are displacing the second harmonics valley to peak relationships. This changes your perception.
Nelson Pass' research shows that coinciding peaks yield a more forward sound, while peak-to-valley second harmonics produces a more recessed sound but wider soundstage.
So, it turns out that, for amps that have predominant second order harmonics, absolute phase IS actually important.
The interesting thing is that you could play with both, depending on how you would like your sound.
Hope this makes any sense, best regards,