The Crimson Troupe are strange, even by Terra's standard.
The mundane explanation is that they're a bunch of assassins pretending to be a wandering actor's guild, a remnant from Gaul after their empire fell, but here's where it gets weird.
Their ordinary mooks share the same game mechanics as the Seaborn. They deal nervous impairment, which is basically sanity damage.
The place where their HQ is supposed to be doesn't exist. There are dead Gaulic Emperors debating each other inside their castle, living people who've been turned into statues, a giant mechanical teddy bear that's made out of clockwork and the soul of a unfortunate Ursus (Big Sad Lock), eternal balls where people have gone insane dancing forever, and a lot of other freaky stuff that wouldn't be out of place in the City.
Oh, they're like the Ring, then.
For context: The City's 'criminal' elements (in quotation marks since everyone follows the Head's rules, even people who'd be criminals in other worlds) are run by the Five Fingers, each of which is a massive Syndicate with power and influence similar to the Wings themselves. We've met three of them thus far: The Index are those weird guys who follow the Prescripts, and the Thumb are mafiosos whose thing is absolute obedience to authority (if Capone was in the Thumb, he'd get shot the moment he questioned his Father's orders).
The Ring was mostly present in Leviathan. We don't know much about their internal structure, but the little we've seen shows that they're a group of mad artists. They have control over some labyrinthine extradimensional 'corridor' that they can connect to anywhere in the City by building an entrance to it.
They hold a bunch of art exams wherein people pose as statues and get killed if they break character, or where people submit paintings, and if their painting is subpar, the painter gets killed on the spot (apparently, art made when one's life is on the line has a charm to it). I'm guessing it's the same kind of thing as the Index making people follow Prescripts. In Ring territory, you are either a good artist or you're dead.
We're also shown an art auction where people have to wear veils and bows before the bidding starts. The items on sale are stuff like "an urn hair collected from every villager in a now-gone town in a nest," "a man who has been dancing in a display case for 8 years now, with time flowing 8 times slower within a said display case," and "a painting, taken from the consciousness of the sole survivor of a massacre to immortalize the carnage."
The Troupe gives the same vibes as them.