Since Hircine is the Father of Manbeasts, it seems appropriate that he'd have something to do with all the intermediate forms, such as hagravens and harpies

However, the Argonians and Khajiit aren't his, and nor are Winged Twilights, Spider Daedra etc. So what are the "rules", if any? Does it need to be a transformation that can go both ways?

And if not Hircine, then do the hagravens have any Daedric affiliation?

I haven't quite figured out haggies and harpies yet. In general though, I view the Aedra as very good at creating but having little interest in managing their creations, and not very good at change when they do attempt it. I view Daedra as lousy at creating anything new but very skilled at manipulating and changing the creations of the Aedra - blending women and spiders to make spider daedra as only one example. And the Daedra, of course, are quite keen on actively screwing with worldly affairs.

Khajiit always thought the Hagravens were just witches that became these creatures through some dark magic (perhaps a "boon" given to them by doing certain rituals for Hircine). In this case, it is a transformation (though seemingly permanent) and thus would fall under Hircines sphere. Harpies on the other hand seem to be born this way. Perhaps they were created by Kyne, or maybe one of the Daedric Princes like Molag Bal or Azura.

So basically:
If it's an individual who under goes an animalistic transformation, this falls to Hircine. If it was/is a change that effects an entire population, then this was not Hircine.

That's a reasonable divide. Person<->animal is Hircine's job. Permanent "hybrids" aren't.

I read a good short story by Larry Niven where a wizard tries to dispel the magic that sustains a werewolf's form, only to find that this was one of the rare instances of a wolf transforming into a human.

Lol, Khajiit would've hated to be there when the wizard realized his mistake!

The morphing between man and beast seems always associated with Hircine. His quest in Oblivion had to do with hunting a unicorn protected by half-man half-bull men. Harpies are found on his realm on ESO.

Now on Hagravens, well, yes they are associated with Hircine and revere him as one of the "Old Gods" much like some Reachman clan revere Namira or Molag Bal. We encounter hagravens who once were witches of Glenmoril in Solstheim. They were probably seduced by power and longevity and thus turned themselves into hagravens. While the Glenmoril Wyrd worship and respect Hircine in nature and offer cures, Hagravens and the Reachmen gravitate toward Hircine's more feral side. The simplistic yet untamed animal savagery.

Hagravens seem to worship more "evil" Daedric Princes.

There are tons of ES material that links Hagravens to Hircine.

Consider also that in Skyrim, Hircine seems to enjoy the hunter becoming the hunted, which fits the lycanthropes he created very well.

I think it's also worth mentioning vampires. Both vampirism and lycanthropy are diseases that can be imbued by the Daedra Lord that created them, gifted from one person to another, or simply inflicted upon a victim. These similarities suggest something of a hallmark of daedric handiwork. Becoming a hagraven on the other hand is a ritual, seemingly setting it apart from other Daedric transformations.