The first, and best-known, of the manuals on the cotillon published in London seems to have been Gallini’s. His A New Collection of Forty-Four Cotillons, appended to his Critical Observations on the Art of Dancing, appeared around 1765. Most of the book is taken up with music and written instructions for the cotillons themselves, but Gallini begins with ‘General Rules’. These aren’t as helpful as they might be since he assumes that would-be dancers are already familiar with the square formation and the numbering of couples around the set. (I write here as a relative newcomer myself to this dance).
He begins by explaining that every cotillon begins with a Grand Rond and that any of another 8 changes may be danced after the figure. Gallini assumes that his readers know the basic structure of the cotillon. He then lists and explains a number of figures and steps – but ‘only those which are used in the following Cotillons’. These are the ones he includes.
Allemande; Assemblé; Balancé; Chaines; Chassé; Contretems;
Moulinet; Pirouette; Poussette; Course or Promenade; Quarrés;
Queue du Chat; Ronds; Rigaudon
It is not surprising that the terminology is entirely French. Indeed, the ‘Frenchness’ of this dance probably added to its appeal in London.
In his instructions for each cotillon (all of which have appealing French titles), Gallini specifies only the opening Grand Rond and then describes the Figure. He does explain the musical structure. In some cotillons, he specifies the use of minuet steps. Some knowledge and interpretation is needed to actually perform these dances.
Dance in History 
Thanks for this series of articles, they’re most interesting!
For what it’s worth, a bunch of the Gallini Cotillions have been animated here: http://regencydances.org/dancelist.php?wAuthority .
Many thanks for your comment. I took a quick look at the animation. I have seen this website before but it was good to have a reminder. I leave it to others to assess it.