kg21: Well, just imagine your typical sine wave graph. Now imagine the same graph mirrored at the x-axis, i.e. with the phase rotated by 180 °. That's exactly what a phase inversion switch does. The same inversion/180 ° phase shift would happen if you connected passive speakers to an amp plus to minus and minus to plus.
And why would one want to have a phase inversion switch? Well, in theory, there should be no audible difference between a signal and its inversion - but for some people it seems to make an audible difference despite all theory, and unfortunately there are quite a few pieces of pro and consumer audio equipment out there that aren't true to the phase and will rather produce inverted output. So, basically, if you don't check and measure both the recording and the playback chain, you cannot be sure, whether the result is inverted or not - however, if you have a toggle switch, you can simply choose what sounds better to you...
Greetings from Munich!
Manfred / lini