Should the bass notes be on the left and the high notes on the right, or the other way round?
If you sit at a piano keyboard, then the bass notes are on your left and the high notes are on your right, so it would seem that this would be the correct way to record the instrument. Pan the bass notes left and the high notes right.
But what if you are a member of the audience at a piano recital or concert? The piano will be sideways-on, with the player at the left. Since the bass notes have longer strings, their sound comes from the full length of the instrument. The high notes have shorter strings and thus are heard mainly towards the keyboard end. Therefore the high notes come from the left, and the bass notes extend all the way to the right, the reverse of what the player hears.
Since the notes of the piano keyboard extend all the way from left to right in even semitone increments, it may seem that the sound from the strings should be the same way in the stereo sound stage from left to right. So the lowest note A is all the way to the left; the highest note C is all the way to the right; Middle C is in the middle and every other note is positioned in proportion.
This however doesn't take account of the fact that in nearly all pianos, the strings cross over to save length (in a grand) or height (in an upright). It is a very rare piano where the strings are all parallel.
So when you sit down to play the piano, the low notes are 'left-ish' and the high notes are 'right-ish', but there isn't any precise sense of positioning. There is however an even spread from left to right, and this is something that should be captured in a recording.
To pan stereo mics from the player's perspective all that is necessary is to pan the mic that is pointing to, or closer to, the bass end of the piano to the left. Pan the other mic to the right. If the piano is to be mixed in with other instruments then you probably don't want it to sound as wide as the distance between the speakers, so pan the channels inwards to get the width as you want it.
In the concert hall, only people in the front couple of rows hear any directional information from the piano at all. Any further back and the direct sound from the instrument is virtually a point source. All of the stereo information comes from the ambience of the auditorium.
So you might consider this if you want your recording of piano to sound like a Beethoven sonata. The close mics should be panned almost center and the ambient mics panned hard-left and hard-right. If you're not using ambient mics, then you will need to add a very natural-sounding reverb. (I should add that panning a stereo pair inwards works better with coincident than spaced mics due to phase issues, so you might want to bear that in mind.)
Sometimes the nature of the equipment we use leads us to create a sound that doesn't correspond all that closely to what would be heard in reality. It is always useful to think about what would be heard naturally without the mics, then aim to mimic that sound in the finished recording.
Great home recording starts with a great home recording studio. It doesn't need to be expensive if you know how to select the right equipment for your needs.