Recently I discussed the difference between unit intervals and integers. Unit intervals are kind of the complement or the opposite of integers, they represent the whole space between each integer, and you could say that they represent the full transition between each consecutive integer.
This is relevant because I was recently reading about the spin group Spin(n) and the special orthogonal group SO(n). The special orthogonal group can be thought of as an n dimensional rotation matrix and the spin groups are spinors which kind of represent rotations as well, but have the confusing feature that you need to travel 720 degrees to return to the same spot. The 3d spinor is the same as the quaternion.
Spinors are confusing, as illustrated by Michael Atiya (from the Wikipedia article on spinors):
"No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of -1 took centuries, the same might be true of spinors."
It dawned on me that spinors are to the rotation matrices what intervals are to the integers. Spinors are the full set of configurations between two rotation matrices. So they do not represent attitudes (orientations) themselves.
Looking at the 3d example, the attitude of an object is its 3x3 rotation matrix. If you attach that object by many elastic strings to a reference frame (like to the walls of the room that it is in) then the 3d spinor (which is a quaternion) represents the configuration of the strings (which do only repeat after a rotation of 720 degrees of the object).
So in this example the rotation matrices are the object and the room, and the quaternion is the strings. It is a full path (or interval) between two rotations, which is quite different to a relative rotation matrix.
To rotate a vector v by a spinnor s, you do: s' * v * s. Same for rotating a rotation matrix by s.
The reason for the double operation (which also explains the double cover) is easier to see if you convert back to the integer and interval analogy:
Integers are just points, which really convey no size. In order to add 1 to an integer, first you must append a unit interval from the integer (think of laying down a metre rule from that point), then you must append the point to the other end of the unit interval, so you are doing the first operation kind of in reverse.
In fact, for any vector you can move to a different vector position by laying down a straight line (an interval) from the first point, then placing the new vector at the other end of the line.
This two-stage symmetric operation can be defined as s' * v * s.
So in fact, basic arithmetic and vector operations can actually be done using these two complementary objects (the point and the interval), we just don't usually think of it that way as we have a simpler system of adding only vectors. But this two object system is exactly what is required for operations acting on orientations.