I did lots and lots of gymanstics with no stirrups and no reins; and also with closed eyes, and I did similiar exercises with students.
We has a lot of fun with the no hands - I would have them do hands on head, hands on shoulders, hands on hips, airplane arms, etc.
The craziest version of this that I ever did (NOT RECOMMENDED) was an instructor back in the early 70's who had the class pass their helmets around their backs over each element of the gymnastic. This was back when helmets were still mostly decorative and didn't have safety harnesses. The interesting thing is that it worked very well. Having to really *think* about passing the helmet completely prevented riders from stiffening/anticipating the jumps.