In mustaches, at least, to twirl means he gave it a twist. You can twirl it out straight, long, if you wished (with some wax, of course).
To curl means he gave it a kind of smile. It wasn't necessarily twisted, just reformed into a C-shape up each side of his face.
You don't wind a moustache however. You can wind a clock, or an electric cord around the base of the iron, for example. To wind something is to continue to wrap it around one solid object (in the case of the clock, you are tightening the mainspring, which is anchored at one point in the centre; you are winding the spring around that single point.
If the mustache were incredibly long, you might wind it around your neck.
In the example below, he clearly twisted it first, to give it stability, and then curved it upwards a bit, just for flair. I personally would indeed wind it around his neck, if I had the chance.
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