To be clear about polarity inversion and phase shift.
Everyone is taking the logic of what I wrote and working backwards, which isn't a valid reading of what I intended.
I noted that polarity inversion is often called a phase inversion because sin(x + 180º) = - sin(x). I railed against polarity inversion being called phase inversion because it causes confusion. Now everybody is confused.
Confused because they think I said that messing about inverting the phase of components of a signal is inaudible, or meaningless.
I said: polarity inversion is often called phase inversion, and I think this is a bad thing as it it causes confusion, especially as polarity inversion is generally inaudible. This is especially so in conversations about phase in general.
Some posers think I said, inverting the phase of a component of the signal is inaudible. I didn't.
If you invert the phase of a component of the signal you did not invert the polarity of the signal.
All evidence seems to show that unless you reach the point of asymmetric non-linearities in your audio chain, absolute polarity is inaudible. Get loud enough and you will eventually start to distort the air, and air is not symmetrical. Loud bass can reach the point where the air in loudspeakers is ceasing to be linear, and causing distortion. It is important to distinguish between non-linearities in the reproduction chain changing the sound from absolute phase itself.
Also, if your signal cannot be decomposed with a Fourier transform it isn't periodic. If it isn't periodic the word phase is not defined. You might be talking about moving the shape of the signal about in some sense, maybe with timing offsets but unless you have a decomposition into periodic components the word phase has no meaning. We can argue about whether your signal has a Fourier transform, but be clear, the definition requires it to be periodic. (To be really pendantic, a Fourier Transform does not require that the basis functions be sin and cos. They do need to be periodic.)