I deleted a longer post about phase, polarity, and delay in the context of loudspeaker crossovers. Mating the sound output from two drivers together in typical loudspeakers is at best is an imperfect exercise. Use of phase shift, polarity flips, and delay (when available) are mainly to prevent very audible suck-outs from deep cancellation. For a while the odd order crossovers were popular because 3 dB bumps were less bad, than deep notches. (but crossover performance also involves off axis lobing, and total spectral energy delivered into the room, etc).
In the context of crossovers a polarity flip "could" improve mating in the crossover region, while resulting in entire bandpasses being opposite polarity from each other. (I am not making a statement about the audibility of that, for some situations apparently a lesser evil).
Try not to think about this too much (i don't). This is why I defer to the speaker design engineers to design speakers. Crossover design is IMO part of speaker design. Back in the old days I actually designed and sold rack mount analog crossovers, but those were simpler times. Now we have better solutions available.
JR
