Can I just correct a bit of a misconception about phase and phase inverting when recording.
Absolute phase can be switched on a desk, but is also affected by the majority of EQ applied in a studio - the very act of shaping the frequency response will alter phase, and this shift will be frequency dependent.
In addition, any vaguely complicated instrument will typically be a combination of overhead mics and a close mic - drums for example. The different path lengths to the different microphones produces a relative phase shift, so even if the mics are all 'in phase' with each other, they will still produce some degree of comb filtering due to path length differences when mixed together. Overheads aren't panned hard left/right, but are mixed into the stereo image, so you have issues with comb filtering to deal with. Remember, a snare hit has energy right up into the 10Khz region, and that's what, 3cm as a wavelength? If the path to the overheads differs by 1.5cm they will be completely out of phase, but at 5Khz, they would be 90 degrees out, and a 2.5Khz, only 45 degrees out. See the problem?
BTW, it's quite common for the overheads to sway around whilst the drummer is working, and by more than a cm ;-)
So, add in different EQ, dynamics processors such as limiters, gates, compressors, EQ on the desk channels all adding frequency, and possibly amplitude dependent phase shifts (so the shift alters as the note decays), and try and explain how the engineer is going to give the correct absolute phase for both the snare, and the floor tom at the same time.
The phase invert is handy for close mics which are on opposite sides of a drum, or for example, on a kick drum (where it's placed opposite the hit side) unlike, say the snare (same side).
For extra points, consider where you would place a pair of mics to record a piano in phase across all of it's frequency range (20Hz to 4Khz).