"Properly" naming rotations of unusual scales?
Consider the following musical scale, with a root of A:
A — B — C — D — E — F — G# — A
This is, quite clearly, A Harmonic Minor.
Let me rewrite the above scale in terms of tone (step) differences:
W — H — W — W — H — A — H
where A refers to an augmented second, or three half-tones.
Consider the following rotation by a fourth:
W — H — A — H — W — H — W
and correspondingly the original scale beginning on D:
D — E — F — G# — A — B — C — D
What would this scale be called?
Consider the regular seven church modes — Aeolian, Locrian, Ionian, Dorian, Phrygian, Lydian, Myxolydian — where A minor would properly be called A Aeolian and is a rotation of C major — rather, C Ionian — and, relevant to the above example, D Dorian.
Just like adding a sharp seventh converts an Aeolian scale to a Harmonic Minor, is there musical nomenclature that a sharp fourth converts a Dorian scale to a Harmonic Dorian, and likewise for the other five rotations?