At the turning of the twentieth century, St. Louis dancing master Jacob Mahler contacted his colleagues across the United States to assemble a set of original cotillion figures -- not in the sense of the 18th-century French square dance, but in that of the party-games-with-dancing, some quite silly, which by the end of the nineteenth century had metastasized into events of their own, separate from formal balls. Using these figures, Mahler assembled a small handbook: Original Cotillion Figures (St. Louis, 1900). The very first figure given, courtesy of Walter L. Curtis of Utica, New York, was called "Stars and Stripes" and seems particularly timely as America celebrates Independence Day.
This is a geometric figure with a mixer element rather than a silly one, but it is a "property" figure, meaning that it requires props, in this case small American flags. It's not clear exactly how many are used; it could be either one per dancer or one per couple. If one per couple, the conductor of the cotillion should decide which gender retains the flag when the couples separate and reunite in different pairs.
The original figure is for thirty-two dancers. I'll go ahead and describe the figure for thirty-two, but the number of dancers could be adjusted, provided it remains divisible by four. The music should be John Philip Sousa's famous march, "The Stars and Stripes Forever". Sheet music and recordings of this piece are easy to find. The base couple dance for this figure is the two-step.
1. Eight couples, having acquired their flags, two-step around the room.
2. At the conductor's signal, they separate and each seek another partner, with whom they continue to two-step.
3. At the next signal, four couples form a column of couples in each corner of the room.
4. At the next signal, the columns of couples march on the diagonal toward the center. When they meet, they separate, gentlemen turning left and ladies right at 45-degree angles, and march toward the walls side-by-side with a new partner.
5. Upon reaching the wall, take this new partner and two-step.
This really is a neat little figure. The dancers form a sort of asterisk, coming in to the center in the form of an "x" and going out to the wall in the form of a cross. If the couples arrange themselves suitably in the corners, each person dances with two or three different partners during the course of the figure.
One doesn't have to wait for Independence Day to do this figure, but it would certainly be extremely appropriate for a July ball or any event with a patriotic theme.