The anti-twister is an interesting mechanism that gives a physical interpretation for a physical state that returns to the same state after the inner part turns 720 degrees.
This has connections with the quaternion sandwich product as the qvq^-1 is similar to the RTR^-1 transformation of the anti-twister.
One thing missing in the analogy is that the quaternion sandwich product can be applied in two ways. The quaternion is an isoclinic rotation, so it rotates by equal magnitude angles on two orthogonal planes in 4D. To create the double-cover of the 3D rotation one of these rotations is cancelled out in the sandwich product. The alternative sandwich product (using an alternative multiplicatino table, or possible using the complement of q) has the cancelled out rotation having the opposite sign.
The anti-twister's twist matrix T rotates around the y axis by an angle y in radius that depends on the radius x. The usual function is a smooth step from y=pi at x=0 down to y=0 at say x=2.
We can create two different anti-twisters if we make the object being rotated 720 degrees not the centre, but a unit sphere. Both the centre and the distance space is unrotated. The two types depend on which direction we rotate the inner space relative to the outer space.
If we rotate in the same direction we get this anti-twister:
