[Edit, 1/14/2013: Newcomers, you may want to see my tag for the circassienne, which will show you all of the Galerie des Modes circassienne plates I've translated so far. They're the same plates as below, but they have longer descriptions that I've also translated, that give the conclusions I took ages to get to here.]
So. The circassienne. According to Wiktionnaire, Circassia was a region of the Caucasus, on the coast.
Young lady in a circassienne trimmed with blonde lace, decorated with tiger ribbon, coiffed with a gallant hat with a loose and braided chignon.
Young lady in circassienne of Italian gauze with a skirt made of the same gauze; the furbelow is decorated with a colored ribbon; she is coiffed with a pouf with a kerchief trimmed with pearls with a feather on the left side, in the Asiatic fashion.
Circassienne of taffeta banded with ribbons on the skirt in another color decorated with gauze in little round leats, and decorated with three large bands of ribbons in different colors.
Circassienne fond de couleur, à bandes d'étoffe peinte et une petite bande de gaze plissée autour: la juppe est d'une autre couleur, et la garniture de la juppe pareille à celle de la robe. La coeffure est un chapeau à la Grenade.
Circassienne in background color, with bands of painted fabric and a little band of pleated gauze around it: the skirt is another color, and the trimming of the petticoat is the same as that of the gown. The headdress is a hat à la Grenade.
Cette Robe à la Circassienne d'un nouveau gout, est de gaze couleur de soufre, la garniture de gaze lilas tendre; le grand falbala et le bandeau qui règne dans la garniture est la même gaze que la robe, le fond des sabot aussi; il n'y a que les bandes de garnitures à tuyaux qui soient lilas, les rubans lilas, même celui de la coëffure.
This circassienne, in a new taste, is in saffron-colored gauze, the trimmings of soft lilac; the big furbelow and the band that reigns in the trimming is of the same gauze as the gown, the bottom of the [shoe?] as well: there are no bands of the trimming that are [piped?] with lilac, the lilac ribbons are also in the hair.
Robe à la Circassienne, la garniture en platitude bordée de gaze; coëffure ceinte d'un ruban garni de perles, terminé d'un noeud d'où pendent des glands: le dessus de la tête orné de fleurs.
Circassienne, the trimming a flat border of gauze; coiffure belted with a pearl-trimmed ribbon, ending in a knot with a hanging tassel; the top of the head decorated with flowers.
Circassienne garnie de Gaze en pouf, la Coëfure un Hérisson ceint d'un Ruban boiteux avec une Rosette sur le côté gauche.
Circassienne trimmed with poufs of gauze, the hair coiffed in a Hérisson belted with a [flawed?] ribbon on the left side.
Jeune Actrice Bourgeoise étudiant son rôle: elle est vetue d'une Circassienne, le juppon garni d'une bande de couleur de la robe: toutes les garnitures bordées de blonde, un volant de gaze au bas de la bande du juppon: les manches garnies de gaze.
Young bourgeois actress studying her part: she is dress in a circassienne, the petticoat trimmed with a band the color of the gown: all the trimmings edged with blonde lace, a gauze flounce on the bottom of the band on the petticoat: the sleeves trimmed in gauze.
Non-period sources: to be honest, I don't trust uncited out-of-period sources further than I can throw them, generally, but sometimes they can yield interesting possibilities.
This is from the NYPL, and dates to the 1880s. I'm honestly not sure whether or not to take it into consideration, as it seems much more like a turque (or something else, I'm not entirely sure as I haven't done that yet!) to me.
Also from the NYPL, but from the 1860s (which seems very odd given the typeface and bold coloring, but I'll accept it). Now, setting aside the fact that the artist cobbled together bits of different styles, which I'm not entirely sure is kosher, the sleeves here are specifically singled out as circassienne. This seems more credible to me, given that he's noticed that that kind of back is specific to the polonaise.
A Short Glossary of Clothing Terms describes the circassienne as "variation of a polonaise: short funnel sleeves, through which fitted sleeves of undergarment drawn through."
In Eighteenth-century French fashion plates in full color, Stella Blum says that the lévite and the circassienne have the same cap sleeves. (Page x.) But this seems to be the only mention of the circassienne in the whole book. In Fashion in the French Revolution, Aileen Ribeiro correctly notes that it's a version of the polonaise, and says that it's not only short sleeves but "'oriental' trimmings such as fur and tassels" that distinguish it.
Conclusions: In general, the styles referred to in fashion magazines aren't governed by enforced rules - different fashion plates can contradict each other. Who knows whether 18th century women were annoyed by this or had their own standards for determining which styles were which! But it also seems possible that sometimes artists and label-writers could get mixed up and draw a turque when asked for a circassienne or mislabel a picture. I strongly feel that that happened a couple of times above.
At the very least, a circassienne-style dress had to have the same back and skirt as a polonaise, trimmed all around the skirt. The sleeves seem to have been different as well in the majority of these fashion plates - of the ones that don't have ordinary sleeves, the plates seem to be evenly divided between short mancherons over longer, more fitted sleeves and full sleeves held in with bands, sometimes in two parts (an upper sleeve and a lower, white sleeve).
I tend to agree with Ribeiro that the "oriental trimmings" were important to the definition. Nearly all of the plates above show at least one tassel somewhere on the bodice or the skirt. (I don't think I've ever disagreed with Aileen Ribeiro.) The trimmings in general on these plates appear to be fussier, in my opinion, than those on the polonaises. "Oriental", in the period, can mean quite a lot of different things, and I wouldn't be surprised if those applied flowers and such were considered exotic.
(Fashion plates thanks to Dames à la Mode, Titam, and 18th Century Blog!)