Bob Dalsemer and Larry Edelman have studied and written about the peculiar quadrille style square dance done traditionally in Independence, Pennsylvania. It is called "The Independence Lancers". CDSS published an article by Bob on this dance in the 1980s but I don't have access to it so I can only tell you what I remember from Larry Edelman's workshop 22 years ago.
The dance is old, pretty unusual in this country. Typical of quadrilles, it has five figures, each figure with its own melody. Its like five short square dances done in a row. The band plays for one figure, pauses for a few seconds, then commences the next tune. One after another. The dance is done without calls. Very nice, a real folk dance passed down by generations.
The following three pages are from the syllabus of the 1988 Camp Kiwanilong in Warrenton, Oregon. Larry Edelman presented the Lancers and other square dances with music by Greg and Jere Canote.