Viviani's Curve approximates the motions of a 21st Century exercise/movement pattern: the Rope Flow Dragon Roll. You can see a demonstration of the Dragon Roll in the first 25 seconds of this video. The mid-line of the rope follows the curve. One simply re-orients the 3D curve so that the "cylinder" of Viviani's curve is lying horizontally on the X-Y plane. Here is the curve drawn with a dot animated to follow the path. Note: the cuboid in this drawing represents the Rope-Flow person. It is present merely as an aid for orienting the curve in a way different than it is ordinarily represented and has no geometric significance. As represented by cuboid, the rope criss-crosses over the person's head.

In a Wolfram U course, I learned that such closed curves have a torque. Ropes used for Rope Flow have a fair amount of torsional stiffness; rope-flowers sense the inherent torque in the curve. Torque plays an important role in driving the rope and controlling transition movements. Have you run across any discussion of the torque of Viviani's Curve?