That said, phase isn’t described in terms of units of delay. It’s described in terms of degrees of rotation for a given moment irrespective of time. The problem with breaking it down as a function of time shift is that time shift would be different for every frequency.It isn’t a "problem" to break it down as a different time shift for different frequencies, it is the reality and why it causes distortion. While I agree that it is more common to talk about degrees of shift, the degrees don’t adequately describe what is actually happening. If I remember my theory, the cutoff frequency is shifted 45 degrees in a one pole high pass filter. Frequencies below the cutoff are shifted more and those above are shifted less. This shift in time, more for some and less for others, results in a form of distortion
BTW we usually describe signals in terms of the time domain (what is the voltage ) or the frequency domain (what frequencies are present) . For instance.. a perfect square wave in the time domain is described by the voltage at any point in time, what you will see on an oscilloscope. In the frequency domain it is described as consisting of a fundamental frequency and an infinite series of odd harmonics of ever decreasing amplitudes, what you see on a spectrum analyzer. If you apply a square wave to a crossover what comes out is no longer square because the different frequencies are shifted in time by different amounts. They are also attenuated by different amounts which along with the change in phase creates something other than a square wave.
Never heard of a polarity domain.