This paper was inspired by a long-time fascination as performer, choreographer, and audience member with swirling movements in dance. The presentation begins by looking at what happens when we rotate our limbs in very simple movements, and progresses to examining more complex swirling movements. We start with a simple gimmick with the arms that has surprising connections to classical dance forms such as ballet. Swirling movements often embody double rotations of the limbs similar to the Dirac plate or belt trick, in which an object attached to a stationary body by a flexible cord returns to its original state after a rotation of not 360° but 720°. This phenomenon is also seen in the Balinese candle dance, baton twirling, poi, and other performance forms. It is efficiently modeled by the quaternions and actually illustrates the mathematical theorem that the group SU(2) double covers the rotation group SO(3). We will look at how this plays out in dance and other performing arts, give some suggestions for simple and enjoyable movement tasks that illustrate the concepts, and see how comprehending the embodiment of the quaternions helps us better understand both the mathematics and the relevant movement arts. The paper ends with an attempt to better understand movements with the simple prop of a piece of paper in a series of dances and workshops created by the author and Erik Stern in the 1990s - and with suggestions as to how readers might have fun creating their own dance movements with paper.