In Scottish it's called a Polite turn, and the phrase has drifted into English dancing in North America, though I've never heard it in England.
Graham Christian says:
There is no evidence of what we call the polite turn at least up through Wilson in the 1810s. I can check my quadrille sources later today, but I'm doubtful that it will be there either, since the quadrille dates to Wilson's lifetime. So, very unlikely to date from prior to 1820 — whether it is an artifact of the 19th century that was then adopted by Miss Milligan is the question. Think of it this way: the polite turn is a way of guaranteeing that both dancers face into the set after a turn. If you're well trained, in 1710 or 1810, you know already to do that (where else would you face?).
Simone Verheyen says:
Here in Belgium and Holland a courtesy turn is what you do as the last part of a ladies' chain. A polite turn, on the contrary, is when you do the 2nd or the 4th change of a circular hey (or square hey) you keep left hands joined with the person you passed, so that you keep “facing each other” first and then face the middle of the set. I think this is Pat Shaw's heritage, he taught dances in Holland for years….. I may be wrong, but that's how I learned it.
I don't understand Simone's distinction, unless she means that the difference is the man putting his right arm round the woman's waist. But that wasn't done in the quadrille, which is where the ladies' chain comes from — it was an open left-hand turn originally. In my opinion the man doesn't usually do a courtesy turn at the end of two, three or four changes — I talk about this in my page on “The Hey”. It's actually the same as the “Square Thru” of Modern Western Square Dancing except that he probably finishes by facing in — but not always — he faces the way he needs to go for the next move. For instance in my interpretation of “Lilli Burlero” the ones finish the three changes facing down and go straight into their lead down through their new twos. You certainly don't want a courtesy turn in Sharp's version of the third figure of “Parsons Farewell”!