[quote author="MikoKensington"]I think the key is the term inversion. Polarity and Phase. At that point the two are the same. Or are they?[/quote]
No, not for our purposes.
For the strict mathematical analysis of infinite waveforms, phase inversion (as described on the link you posted, Miko - I've never seen the term used in any formal text before) and polarity inversion of a periodical symmetrical waveform will give the same result.
In audio, however, even periodic, symmetrical signals (sine waves, for example) are discrete in time, i.e., they have a beginning and an end; so if you phase shift an audio signal 180 degrees, it will be time delayed by 1/f seconds. Try to use it in a balanced system and the very first and the very last cycles will still come through loud and clear.
Also, remember that a phase shift of 180 degrees at one frequency, can be 3 degrees, or 3000 degrees at other frequencies. So the very concept of a 180 degree phase shift of a harmonically complex waveform doesn't make sense at all!