When something sounds off, really. Sometimes I'll be listening to a new LP (or old tape) or one I haven't thrown on in years and it will sound wrong. Then I flip the Phase Switch and (sometimes, Presto!) it sounds right.
I had one for about 10 years with my Melos 333 preamp until it croaked. It was easy to check proper phase for a record/CD.
My stereo amp (Antique Sound Lab Tulip 2A3 SET) had a switch to invert the polarity of one channel, so you could bridge it to be a monoblock. I had a friend add another switch, so I can flip the polarity in both channels easily. For me, having an easy way to switch polarity on the fly is an essential feature. I listen to determine the "correct" (best sounding) polarity for a particular CD (sometimes individual tracks on a compilation album) and mark it. I have found that my collection is pretty evenly distributed polarity-wise. If I didn't have a way to switch, half of my music would not sound as good as it could. Incorporating a polarity switch into a preamp or amp is relatively easy if it has transformers and/or is balanced. Otherwise, not so much.
Another thing to note is that you may not hear a difference switching polarity if your speakers are not phase coherent (like my Audio Note AN-Es). Some speakers are purposely wired with some of the drivers polarity-inverted!?
In my own experience only a minority of albums,whether LPs or CDs, had a crystal clear difference. Many pop recordings have all kinds of phase anomalies anyway due to the recording process. The most reliable way I found to hear a subtle difference was to concentrate on the relative amount of centerfill of the stereo image. Of course that doesn't work on hard channeled 60s stereo mixes but for more balanced stereo mixes it is a useful marker with in-phase associated with more centerfill.
Depends on if it's inverting the polarity of just one channel 180 degrees (not really useful here as I can count the times I have heard a recording that was out of phase on the fingers of one hand), or is an "absolute polarity" switch, which will flip both L & R. This can be much more useful. On many recordings you're unlikely to hear a difference, but on some it can really make things sound better. E.g. no speaker is fully linear, and if you have that initial 808 kick transient pulling the woofer in, instead of pushing it out, it can make an audible difference. But it can be pretty system dependent.
I have had it. Rarely can I hear a difference. I think this is one of the main reasons: The other is, to many people, it's just not that audible.
I am the same. I have an option on my Cyrus cd player to opt for phase invert. I cant tell the difference