Usually this is only in preamps (especially tube preamps)
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Typically for preamps the number of amplification stages with each inverting the signal determines whether overall it is inverting. Three stages would be inverted. I had a preamp once that had a invert switch. When you pushed it, another tube was in the circuit uninverting the three stage preamp. While there are two and three stage amps, I cannot think of a circuit that would need to have the output inverted.
If you can't tell the difference in sound use a CD with vocals , to hear the difference. If you still can't your speakers aren't setup to image correctly. There are lots of threads on speaker setup to help you out. Also the bass will be kinda sucked out, though keep in mind it is a subtle difference.
Not that simple. Almost half of all vinyl and CDs have the absolute polarity inverted. So you really need a polarity inversion switch on one your components (mine is on the DAC). Some of the labels with reverse polarity are RCA, Mercury Living Presence, MCA, Capital, DG, London and some Sony. So a CD with vocals will not help unless you know the polarity of the CD itself.
Tbg...Tube gain stages invert polarity, but a "cathode follower" (unity gain) which is usually included to provide low output impedance, does not. So your rule should be to count the gain stages.
Yarvis...most test discs deal with "relative polarity" as you call it. This relates to the two channels being in phase with each other, and the out of phase condition is very easy to hear. Absolute polarity, where the two channels are in phase with each other but may both be inverted is much much harder to detect and sometimes impossible. Besides, as Elgordo says, who knows which way is correct?
Elgordo...Sorry if I misunderstood you. I thought you were just pointing out that absolute phase is not maintained in recordings. I'm told that different recordings are different, different cuts on a given recording, and even different tracks in a mixdown. I find the effect hard to detect, and even if I could I don't think I could throw the switch fast enough to keep up with all the confusion!