The can-can itself descends directly from the appropriately 2/4-timed galop, a popular nineteenth-century European country dance which itself would later give rise to the...polka. After a spell as a social dance, the can-can's flashier aspects would lend itself to its development as a performative style, a good demonstration of which is offered in this vintage clip:
Notice the splits, please, as well as Offenbach's "Infernal Galop," the musical selection so synonymous with all things can-can that it's referred to by the dance's name more often than not.
While this video gives us the most classic of classical can-can, of note was the social, coupled nature of its earliest incarnation, with men and women performing the dance's assorted splashy movements in concert. While video of this take on the floor dance is difficult to locate, the general idea, one might guess, lives on rather well in Kharis Ralph and Asher Hill's take on the style, which is preceded by an appropriately bubbly take on Gigi. From 1:54 onward, Hill's movements, from high kicks to even more, match Ralph's to an almost startling degree:
Of course, the traditional French can-can is the centerpiece of Nathalie Pechalat and Fabian Bourzat's "Moulin Rouge" dance (performed here for the Parisian Trophee Eric Bompard audience), wherein the partnered aspect is less about a shared performance and more about a patron's flirtations with a music hall dancer. Traditional skirt choreography, little utilized in Ralph and Hill's more couple-oriented piece, is a major above-the-blade component of this can-can:
As a sidenote, the slowed-down rendition of Offenbach's piece accompanying Pechalat and Bourzat's Yankee Polka sequences is indeed but a slowed-down rendition of the "Infernal Galop." But at least it isn't this.
The audience-grabbing appeal of the can-can notwithstanding, however, another couple demonstrates that there's more than one way to de-Yankeefy the Polka.
The bal-musette might be the main source of inspiration for any French-themed program that tackles a social rhythm like waltz or polka and aesthetically calls back to the late nineteenth through mid-twentieth century. This was the angle taken by a team like Delobel and Schoenfelder in their 2010 Olympics original dance, an interesting little bridge piece between the waltzes and the can-cans. And if the genre reached its peak popularity in the 1940s and '50s, then Alexandra Paul and Mitch Islam's Edith Piaf-scored waltz/polka might be considered the ultimate heydey of bal-musette kind of program -- even if Piaf isn't often associated with social dance.
One irony is the fact that Piaf's waltz "La Foule," which kicks off Paul and Islam's SD, is actually a cover of Peruvian vals "Que nadie sepa mi sufrir." Piaf's rendition, though, adjusts the pacing and substitutes in more traditional bal-musette instrumentation, transforming a South American tango cousin into essentially a sprightlier version of French musical waltz (compare with "Padam Padam" or Madison Chock and Greg Zuerlein's 2010-11 use of such).
What makes for a sprightly French waltz song, however, is a little different from what typically may accompany a French valse dance. YouTube's pickings are slim, but a few moments with the valse musette or the folk waltz are reasonably instructive:
Here we come again to the interesting quandary of competitive skate versus traditional floor dance, and the valse musette as demonstrated in Jazz Age-era clips is possibly even less compatible with the demands of an ice dance program. The rotations at 0:56-0:58 and 1:42-1:43 in Paul and Islam's program above might be the closest approximations to valse musette movement, with allowance made for physics and a slightly more laidback tempo. Certainly, the greater part of a short dance is concerned with the inclusion of required elements; while a country-western program, for example, can borrow from that style's vernacular in a variety of ways, the valse musette is, as can be seen, a fairly repetitive dance. Paul and Islam's program holds together tightly -- and is probably a stronger waltz, with its emphasis on closed holds and close skating, than many others calling upon that rhythm -- but its style may be less traditional French dance and more idea of French dance. It's not at all a choreographic parent, but there's a certain theatrical stylistic sensibility just a little reminiscent of, say, a French-set musical.