rival or hedge wizards?

I had a concept of a non-hermetic mystery cult which has developed the ability to work together by initiating (via mystery cult rules) the virtue unaffected by the gift, so they can manage to cooperate without parma magica. How much of a game changer do you think this world be- would they be classified as a rival tradition or just hedge wizards with a curious feature? I haven't gone much farther with this concept, but I would assume being able to work together unaffected by the Gift and with the tight knit fabric of a mystery cult they would have a well developed magic system.

There isn't really a distinction; Rival Magic is just another book of hedge traditions. (To the Order, either you're a magus or a hedgie.)

That aside. To me, the distinction would be "is this a political affiliation of wizards that act in concert toward organizational goals?" If so, they're a rival tradition, because their stories will lead them into contention with the Order. If they can work together as needed, but aren't an Order, Brotherhood, nation, or whatever, then they're a hedge tradition.

Learned Magicians could already do this by granting themselves the virtue via a charm, and Graugrach could do the same with Grant. The introduction of Unaffected by the Gift as a virtue was a significant game-changer for any tradition that could simulate virtues. That said, co-operation is only valuable if co-operation is of significant assistance to all magicians involved. Hermetic Magi have The Wizard's Communion, which permits powerful Aegis of the Hearth to be cast at a low (per magi) cost annually, which permits them to stand up to powerful supernatural threats (Might 30-40) that individual magi probably could not. Other advantages flowed from this (specialization of magi, for example).

Hedge/Rival traditions manage to function without Unaffected by the Gift - primarily because they don't derive huge benefit from working together. Even Learned Magicians - who pass around texts of spell information in the manner of Hermetic Magi - manage fine without close contact between the Gifted members of the tradition. So if a tradition is to bother with the difficulty of initiating Unaffected by the Gift, they should have a compelling reason to work together in the first place (better than 'lab assistance bonus', anyways). The Ceremony Virtue is probably the first place to start with that sort of thing.

It's a good start for a Rival Tradition, but by itself isn't enough. A Rival Tradition is, for me, a tradition that can form a challenge for the Order as a whole. If their magic is weak, it's unlikely that the mere possibility of cooperating will elevate a hedge tradition to this level. However, if they wield powerful magic, alone or in cooperation with others, then being social in this way will allow them to become a Rival Tradition (rather than a diffuse body of powerful individuals). Even moderately powerful magic plus cooperation can form a Rival Tradition, if they are well organized.

It's worth noting that all of the traditions in the Rival Magic book cooperate (except the Muspelli for their own reasons, and they still have a goal) because they have shared political or magical goals. The Amazons and Soqotrans are nations, the Augustans are a political conspiracy as well as an initiation and the Muspelli are here to release the jotnar. Vitkir and Mathematici, while every bit as powerful as the other guys, have no overarching organization or goal - they're just traditions of magic*. Folk Witches organize in covens but nothing beyond that.

That, more than powerful magic, is what makes an Order.

*Excepting the Raudskinna Compact, which came in in Rival Magic and functions as a Rival Order without a single magical tradition.