The Sixdrilles are a clever reworking of the figures French Quadrille (or First Set) for a group of twelve dancers in the form of a square of trios, each consisting of a gentleman and two ladies. I have two Scottish sources for them, which match fairly closely:
The Ball-Room, by Monsieur J. P. Boulogne (Glasgow, 1827).
Lowe's Ball-Conductor and Assembly Guide…Third Edition, by the Messrs. Lowe (Edinburgh, c1830)
Monsieur Boulogne is billed as French, but I know no more about him. The Messrs. Lowe were a group of four brothers, all dance teachers, one of whom eventually became famous as dancing master at Balmoral for the family of Queen Victoria. Their book is difficult to date, especially since it is a third edition. A reference to the Sixdrilles being created around the time of the coronation of Charles X puts it at 1824 or later, and a late reference to the opera Guillaume Tell (Paris, 1829) at the very end of the book suggests 1830 onward. The last half-dozen pages look like a later attachment, however, and may have been added for the second or third edition. The Sixdrilles appear much earlier and are integrated into the overall work.

