By which I mean keep all the same notes in the same order, but start on a note that isn't the root.
Let's work in C major. There's obviously the special case that if you start on the 6th (sub-mediant, same as going down a minor 3rd) you are in the relative minor key - Am here. But what about the other 5 possibilities - does the 'scale' F, G, A, B, C, D, E have any meaning, or B, C, D, E, F, G, A?