I haven't square danced since I was in 5th or 6th grade. I don't recall which grade I was in since I've slept since then, but the way I remember it worked was the boys and girls lined up on opposite sides of the gym and then it was a scramble to "find a partner".
That was not the routine all the time, sometimes the boys picked the girls, and only once I remember did the girls get to pick the boys. I was not one of the "smooth operators" if you know what I mean, so I was never very comfortable with the partner pickin' part of the activity.
So you have to know that the only reason I remember the one time that the girls got to pick the boys was that the prettiest girl in class picked me that day. She may have felt sorry for me for all I know, but it sure made up for all the dance days of discomfort for a young guy trying to figure out who and what he was.
Though I have not square danced in a very long time, I have seen some square dancers perform and they are something to watch! We had friends that were in a square dance club and the dresses and shirts the guys and gals wear are something to behold. A lot of practice goes into square dancing, and a lot of good exercise and socializing can be had as well.
What brings this all to mind is the following video that I was told about. Now around my place, the tractor color is all wrong, but the amount of practice to perform at this level must be "exhausting"...