Mirada - the meeting of the eyes - and the cabeceo - the nod of the head - traditionally, this is how you choose your partner in Tango.
"But this is America!", we hear. "In America we just ask someone to dance. And women don't passively sit around and wait for the man to ask."
Mirada and cabeceo actually give the woman at least as much power in choosing a partner as the man. The woman initiates a mirada from her side - scanning the room to find someone she wants to dance with, meeting his eyes with a steady gaze and a small half-smile. If she does not want to dance with him her eyes bypass him and move on to a more desirable partner. If she meets his eyes, he nods, or tilts his his head toward the dance floor - the cabeceo. She can answer with her own cabeceo - a smile and nod. They have agreed to dance.
This is far from passive on the woman's part. In fact, American women often have to overcome their initial reluctance to be willing to boldly meet a man's eyes.
So how can mirada-cabeceo fit into a typical American milonga? I believe we can make use of the basic principles without necessarily being rigid about it. (Footnote: My preference is for the traditional mirada- cabeco, but I recognize that a lot of communities do not use it, and a lot of Americans do not like it)
I am specifically addressing American milongas. The rules in Argentina are much more specific. Part of the difference is that, traditionally in Buenos Aires, single men and women sit separately, and do not mingle except on the dance floor.
Men - if a woman is talking to someone, or looking down at her cell phone, or sitting rubbing her feet or holding a drink, and not looking expectantly around the room, she probably does not want to dance. So don't ask. That does not have to be the end of it. If you know her, you might say "hi" and add "save me a dance?". To which she can reply " How about this (or the next) tanda?", or the less encouraging "Maybe later". If the latter, don't ask again without some more positive indication that she is ready to dance with you.
The flip side of this for women - If you do want to dance, make it obvious. Look around the room. Sit or stand in an accessible location. Put away the phone.
Most venues in America are OK with women asking men to dance, but women should observe the spirit, at least, of mirada-cabeceo. Make eye contact. If the man ignores you, and won't meet your eyes, is looking down at his cell phone, don't ask him to dance. If he meets your eyes, YOU can make use of the cabeceo - smile, and nod toward the dance floor. Or if you know him, you can yourself say "Save me a dance?"
When you go to a new milonga, observe the customs. If it is clear that a verbal invitation is the norm, go ahead and ask. But make eye contact first if possible. Courtesy dictates that you do not force yourself on someone who does not want to dance with you. Courtesy dictates that you accept a refusal graciously. Mirada- cabeceo represents the most courteous way to accomplish this.