I know of one dance that is quite different from what I am going to talk about here - tango. In tango traditionally you invite using the "Cabeceo". The translation is "nod of the head", and it is a non-verbal invitation to dance.
This is an excerpt from one of my books - "The Secrets of Social Dance - How to Become a Popular Dancer" which you can find on Amazon as ebook and paperback.
Basically, it works like this: The man looks at the woman, and when she meets his gaze, he nods with his head to indicate he wants to dance. If she accepts, she will move towards him, they meet and dance. If she doesn't accept, she will look away. In addition, a similar Cabeceo is also used between leads when entering the dance floor. You "ask" a dancing lead with a cabeceo if it is ok to enter the dance floor. If you get the Cabeceo back, you may enter. Another detail from tango is what is called a "Mirada" from a follow to a lead. In a way, it is like asking someone...
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