Is Phase and polarity the same thing?

If you reverse (interchange) (+) and (-) wires you shift phase by 180 degrees, at all frequencies. That's why "polarity" (which way the wires are connected) is often described as "in phase" or "out of phase".

The problem with "phase" is that the phase is being changed somewhat (that is: not 180, but perhaps depending on the frequency from -90 to +90 or whatever) by many of the bits and pieces of electronic gear. Look at speaker measurements! the phase shifts with the frequency, (because of the crossover etc) the electronics do some of that too. So being able to really tell what the phase is is a miracle of hearing! (and better equipment.)
Rock and Roll recordings usually have more mixed up phase, because of all the console editing etc.
So just because some folks claim they can hear it, does not mean other (who say they cannot) folks can't. It just means the equipment is messing up the phase so much that polarity cannot be determined by their inexperienced ears!
I used to love planar speakers because the polarity was very obvious. Sound from "in front" of the speakers was correct polarity. Sound from 'behind' was reverse polarity.

It is important to distinguish between absolute phase and relative phase. There is no standard in the recording industry for absolute phase , that is why so many records sound better reversed. Relative phase means the speakers being in phase with each other. Perhaps they should be called polarity rather than phase, but no one does.

Elizabeth - In addition to that there are two different definitions of pinout for balanced cable. If amp and pre don't follow the same pinout phase will be reversed.

Planar speakers are nice because they don't create peaks and valleys of the bass with the distance.

Wow, how frequently words fail us. Most test records use the word "phase" to deal only with whether both speakers share the same positive and negative connections throughout all that goes before them. Usually if you experience the signature disconcerting location-less sound, one speaker's connections are wrong, the red connection on the speaker wire is connected to the black binding post.

Polarity involves whether what you get in the drivers moving outward when the microphone at the recording retracted as the sound impacted on it. This would work only were recording engineers meticulous in wiring all microphones the same. Since few are concerned with this, what you hear is unlikely to have similar polarities across all microphones. All you can do is to use your polarity button the way it sounds best on each record.

This is further complicated by how your manufacturer accomplishes polarity change. Most do so by adding another stage of amplification to invert the signal. This usually sounds worse than the preamp without this additional stage.