One lowered sixth
The harmonic major family starts from the harmonic major, 1 2 3 4 5 b6 7: the major scale with its sixth lowered, which the same source also describes as the harmonic minor with its third raised. Its rotations are Dorian b5, Phrygian b4, Lydian b3, Mixolydian b2, Lydian augmented #2 and Locrian bb7, 1 b2 b3 4 b5 b6 bb7.
Names that say where they sit
Five of its seven names point straight at a diatonic neighbour. Dorian b5, Phrygian b4, Lydian b3, Mixolydian b2 and Locrian bb7 are each a church mode with one degree lowered, and the neighbour table confirms every one as a single altered degree. Its modes also reach the melodic minor, harmonic minor and double harmonic families; lowering the harmonic major’s second gives the double harmonic.
One family fewer than its mirror
Each harmonic major mode has three or four neighbours, a few fewer than the harmonic minor modes, because this family touches one family fewer. The Neapolitan minor, which the harmonic minor reaches by lowering its second degree, is not one altered degree from any harmonic major mode: a mirror keeps a family’s interval content, but not the families it happens to sit next to.
Mirror of the harmonic minor
Reflected around its root, every harmonic major mode lands in the harmonic minor family, and every harmonic minor mode lands here, so these two families are mirror images of each other. Distances between notes survive a reflection unchanged, which is why both families have exactly the same interval content although no mode of one is a mode of the other.