The style of basket I first learned is known as Cherokee double-walled basketry. It starts at the center bottom of the basket, goes up the inner wall to the rim, then turns and goes back down the outside. This results in a very sturdy basket made from relatively fine reeds. I had always thought underwater basket weaving was a joke, but the process is a soggy one which tends to restrict it to an outdoor, summer activity.
I sometimes teach a class on this technique.