On a number of occasions over the years I've been aware of a Left - Right signal reversal. Using pop music as a test source, it's impossible to detect this reversal because there are no rules about which instruments should occupy particular positions across the L-R sound stage.
But for classical music, with rare exception, the musicians are arranged in a rigid plan from left to right, just as you would hear them in the concert hall playing a contemporary western symphony or similar*. It beggars belief that a commercial CD could be mastered with L and R transposed, so what is the problem?
It seems that the issue is that many - perhaps most - 3.5mm jack to L - R phono cables are incorrectly wired inside or colour coded outside. There is a perfectly clear standard of how the unbalanced stereo three-way jack plug should be connected and it is the top picture here. The tip (marked 3) should be left channel, the ring (marked 2) right and the body (marked 1) the common ground. But the tip and ring connectors are frequently interchanged.
How can you be sure that the marked left channel really is left? If you buy a fancy expensive interconnect will it be wired correctly? Is this somehow related to cable length? The answer is that, in my experience you should completely disregard the colour coded left and right identifiers on the phono plugs (or sockets). Just don't believe that left is correctly marked as left, and right as right. Ideally test the continuity with a multimeter; the tip should connect to the left channel. And no, price doesn't seem to be an issue, nor brand, nor length. I have been caught out on this a few times and now I always check with a meter and correct the colour markings before using a 3.5mm interconnect cable.
What happens if you don't have a multimeter? You could arrange a bulb and battery. Or you could play classical music and check that the violins are really on the left side. Or you could download this little channel identifier. Here is the embedded clip and here is the downloadable file. Burn it to CD and then you can check your entire system. Here is a brilliant web + video walk through by Sir Simon Rattle of the modern orchestra.
(Clips to follow with pictures).
*Note: the position of the piano (and maybe other soloists) may not follow this general rule if they have a significant individual role in the performance. and/or the composer and/or conductor have their own particular preferences for orchestral seating. But the violins are (as far as I know) always on the left side, as a block.