Depending on how you look at it, blessings from a god imbued in their flesh and armor and weapons to become some of the most dangerous weapons on nirn is pretty super special. Maybe not secret.
Pretty secret, pretty special. How many other warrior organizations are werebeasts?
Okay, that one's a bit of a joke, but their super special secret power is that they biblically know Vivec.
HER HANDS
I was agreeing with Her Hands being pretty damned awesome. They are called the most lethal warriors in Tamriel for good reason. One Hand took down an entire armed group single handedly in one account. Almalexia had set up some rebellious people or something spreading dissent. Then promised them safe passage through the gate if they could get through unmolested. They were prepared to fight an entire garrison of soldiers. They faced one Hand. And they got slaughtered.
COMPANIONS
The companions got nothing. ONLY the circle could turn into werewolves and that was like 6 of them? The other companions did not have that going for them.
BUOYANT ARMIGERS
I really have no clue about them except the origins of the Buoyant armigers. They were rewarded supposedly by Vivec when Vivec was off killing his misbegotten abominable children and they took one of them down. Vivec then started the Order right then and there with the survivors of the encounter. So while they have a legendary origin idk if they got super powers.