re: "Why not just use the non-inverting input in all cases, and avoid the whole question of whether the inputs/outputs are in phase or not?"
That's a good question.
If you are mixing signals and then further downstream comparing them in some way with other signals, then you'd want to preserve the phase.? ("Phase" is not the right term, it's really polarity, but you get the idea.)? As an example, if you mix 4 signals in mixer stage? A, and then mix 4 signals in mixer stage B where one of them is the output of stage A, then you would probably want to preserve the phase so that the inputs to stages A and B don't end up out-of-phase with each other.? That's important if the signals come from microphones, because each mic can pick up some sounds that the other mics get, and having out-of-phase mics on the same stage can be a disaster.
In professional audio, they take care to make sure that polarity (or "phase") is preserved, between input and output.? That way you know that if you add a device into your audio signal chain, it won't mess it up.? It is really bad if you invert the Left channel signal but not the Right channel too.
But the other question is, does absolute phase matter?? If I invert the polarity of a single audio signal, can I hear the difference?? The answer appears to be, yes, to a limited degree.? Many experiments have been done over the years, which show that our ears can hear the difference, in SOME cases.? When you hit a drum, that initial burst is a positive increase in air pressure, and that's how we should hear it.
Having crappo speakers probably just scrambles the phase and then it might not matter so much.? But with a good audio system ("high fidelity"), and good clear sound sources to begin with, and sounds that are familiar to us, it's been shown in blind A-B tests that we can hear the difference.
The advantage of using an inverting amp as a mixer is that there is (ideally) no interaction between the inputs.? In this case, interaction could mean that some of input #1 feeds back into the other input signals, which might be a bad thing for those other signal?sources (e.g., unwanted crosstalk).? Also, if I have a volume control on input #2, as I change that volume control (typically a potentiometer), it changes the impedance feeding input #2 which also affects the levels from input #1 slightly.? That could be a little annoying.? If it's an inverting mixer with a summing point that is a virtual ground, none of that happens.
Andy