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Hah!!! an imposter!!!

It's not fair I tells ya.

Stan,
 
Leave bapü alone. He's may be a poseur with that umlaut but we need to find out if he's legit.
 

Polarity generally has two (opposing) values while phase relates to a segment in time often for a cyclic or repeating action.
 
Obviously, there are times when two phases for a cycle can be exactly opposite of each other (180 degrees and -180 degrees for example) so the cycle can have polarity as well.
 
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Alternatively, a phase is something your kid goes through to drive you crazy until you send him to the Antartic to gain a little polarity in his life.  (Tomorrow we'll discuss bipolar people.) 

bapü 
" delayed by an angular value, which relates directly to a position in the cycle, "
 

Think of it in terms of missionary position. 

The part I am having trouble with is the idea that after you flip polarity, how do you describe the phase?
 
While the statement "a 180* phase shift on a sinusoidal signal is not the same as flipped polarity" makes sense to me, I can not reconcile, or articulate an idea of how you can flip the polarity of a signal and not consider that the phase angle has shifted to compliment the orientation to the baseline.
 
If I remind myself that phase is frequency dependent than there is always a periodic "per time duration" consideration whereas with polarity flips there is no time change involved. Is it that simple? If there is no time involved there is no "angle" to consider a relation to a baseline to?
 
I know I am asking the question in a goofed up way... but it is still a sincere question. :-) 

Polarity is multiplying by -1.
 
Phase shift is a shift in time.

Polarity generally refers to an electrical connection, and is either of two states. In general a reversal of polarity can be considered to be the mirroring of the signal across the zero amplitude line in the amplitude domain. Phase refers to the displacement of the wave along a time line, and is infinitely variable. If you reverse the polarity of your speaker connections the phase is inverted relative to the previous state. If the signal is a sinusoid then it is shifted by one half cycle or 180 degrees. If the signal is noise, the description of the phase shift by radians or degrees is not so obvious.

You can also consider a DC offset sine wave, where everything is above zero. A polarity flip gives you a very different result than 180° of phase shift.