The War of the Quadrilles

Originally there were two dances, both of which were called contredances or quadrilles. French contradances, also known as quadrilles, are the ancestors of the modern square dance, and the english contradance are the familiar longways set dance that is now called Contradance.

They did not always co-exist peacefully:

from http://www.oldtimeherald.org/archive/back_issues/volume-9/9-3/dare-2-b-square.html

The War of the Quadrilles

Another chapter in the history of the contras vs. squares rivalry occurred
two hundred years ago, in 1804, in New Orleans. It has been described by dance historian, Maureen Needham Costonis, as "The War of the Quadrilles." In December 1803, New Orleans became United States territory as part of the Louisiana Purchase. Prior to that time, the favored dances among the city's predominately French-speaking Creoles were French Quadrilles, referred to at that time as "French contredanses." French Quadrilles had not yet come to the United States, where "English contredanses" [contra dances] were still in fashion. On January 8, 1804, only a few weeks after the transfer to the Americans, a quarrel between the Creoles and the Americans broke out at a public ball, over which type of dance should be done. According to one account of the incident: "Two quadrilles, one French [a square], the other English [a contra], were formed at once. An American, taking exception, brandished his stick over a fiddler, and there was at once, great turmoil..." Eventually, after much persuasion, "...the French quadrille [square dance] was allowed to go on, but the American interrupted it on its second time around with an English quadrille [contra dance], taking his position on the floor; some one cried out: "If the women have a drop of French blood in their veins, they will not give in!'"

The contra-square controversy continued and erupted into a brawl at
another public dance later that month. As reported by the New-York Herald on March 10, 1804, the dancing began with cotillions [squares], followed by a country dance [contra] for twelve couples. The musicians were instructed to end the contra after all of the couples had been active, but six additional American couples joined the end of the line and demanded that the music continue. In the ensuing ruckus, fiddles got broken, swords were drawn, and dancers were arrested. In an effort to resolve the issue and let the dances continue, the New Orleans City Council stepped in, prohibiting citizens from carrying arms to the dances, assigning policemen to keep order, and legislating a rotation of dances: two "French contredanses[squares]," followed by one "English contredanse [contra]" limited to twelve couples, and then a waltz, to appease those of Spanish descent.

Following the War of 1812, with the exception of New England, which
remained pro-English, Americans completely abandoned the tradition of
English contra dances in favor of the French quadrilles. Quadrilles were
promoted as being "new" and "fashionable" and contras were seen as
"rustic" and "old-fashioned." While remnants of some of the formal
quadrille figures remain in New England squares as well as western
squares, they had less influence in the South and mid-West, where visiting couple squares became the popular form.