Inverted chords are a thing. But are inverted scales a thing?
What do I mean with that? A scale is defined by a certain set of notes where one of them is the root key. Playing the root key feels like coming home and most melodies in that scale start with that note. So let's say in a song I'm using the notes of a certain scale. However, the note that it feels like coming home to and I often start a melody with, is another note than the root key of the scale I'm using. Could you say I'm using an inverted version of that scale? Is there any other name for it? Or doesn't it really matter and I'm still using the same scale? If that is the case, wouldn't that mean that C Major and A Minor are not two different scales with the same set of notes, but actually the same scale which just has two different names?
An example: For now, let's just assume C Major wouldn't exist as a named scale. Yes, weird, but bear with me. However I'm making a song in a scale that would be C Major. Since it has the same notes and I want to be able to communicate what scale I'm using, I'm would say that my song is in A Minor - second inversion. Of course, it's a bit weird that I'm using a major scale and I'm calling it a minor scale, but I think you understand my point.