Phase directly relates to at what amplitude the waveform is at at any given point in time relative to another driver or reference point and at a given frequency(like speaker A vs speaker B or from when time=0 ). The phase relationship between drivers is going to change over the frequency range and is most critical near the crossover point. Phase is determined by the travel distance from the speakers to your ear, your crossover network, and your enclosure. Different crossover points and slopes cause different phase relationships between drivers. The enclosure and crossover network are the primary driving points of why it sometimes sounds better to invert the phase and not really the distance although the distance changes phase slightly (not as much at low frequencies as it does at higher frequencies).
The reason some amps need to have their phase switched is based on the types of crossovers and their slopes. As you increase the order of the filter (6dB, 12dB, 18dB, 24dB), you change the phase by 90 degrees at the crossover frequency. A butterworth 6dB slope crossover will have a phase shift of 90 degrees at the crossover frequency (Fc). With a 12dB slope, the phase shift will be 180 degrees at Fc. If your mids are crossed over at a different frequency than your sub or if the filter is different, then the phase relationship between the two drivers is going to be off and flipping the sub 180 degrees out of phase could bring that difference down, it could make it worse, or it could change nothing (given a 90* phase shift to begin with as flipping it 180 degrees will still leave your drivers 90* out of phase at the crossover point).
Given an ideal situation where you have two drivers crossed over at the same frequency with the same filters (type and slope), both playing infinite baffle, and having relatively short distance difference between them to your ear, you should get poor results flipping the phase as the two drivers will be very close to in phase with each other at the crossover frequency. If the filters or crossover points are different, then your relative phase at Fc is going to be different and flipping the phase may help.
Also note that the enclosure is going to have a very large impact of the phase response of your subwoofer. Enclosures are filters and an enclosure not properly designed for the specific driver could have a wild and crazy phase response at the crossover frequency. Also note that the higher order enclosure you have, the more rapidly the phase will change.
I hope that clears things up.