But that inverts it around 0DC; i.e. the 0DC centerline is the "mirror" causing the inversion. Yes there is inversion, but if the waveform is asymmetric, there is also DC shift.
Straight phase change, OTOH, whether it's a 180° shift or rotation or an instantaneous inversion, is relative voltage independant; the waveform mirrors itself instead of using a voltage reference as the mirror.
In waveforms that are not symmetric around 0DC, this makes a difference as it does produce different results.
EDIT: I'm about to run out the door for the day, so I'm out of time for now. But in the meantime I ask all the polarity/phase equalitists to consider the asymmetric wave form and it's RMS vs. crest factor value as well as it's energy distribution by frequency (i.e. it's spectral fingerprint), and how those can indeed be potentially affected differently by a polarity inversion versus a phase inversion only. It's not just DC offset that's at stake here; it's potential differences in the actual sonic character of the signal.
G.