Compose two rotations by laying their arcs head to tail on the sphere. The third side of the spherical triangle is the product. Drag the globe to look around.
A turn is an arc. The rotation by φ about the axis n is drawn as an arc of length φ/2 on the great circle perpendicular to n, running counterclockwise as seen from n. An arc can slide along its own great circle without changing the rotation, just as a vector slides along its line.
Composition is head to tail. Slide the first arc (blue) so it ends where the two great circles cross, and slide the second arc (orange) so it starts there. The dashed arc from the start of the first to the end of the second is the product. In quaternions: (S Q̄)(Q P̄) = S P̄.
Order matters. Swapping the order reflects the triangle, and the product changes. That is non-commutativity you can see.
The double cover. An arc is really a unit quaternion q, and the arc for −q is the complementary arc on the same circle. Both give the same rotation. An arc of length 180° is −1, a full 360° rotation, which is Dirac's belt trick.
The torus. When both rotations share an axis, both arcs lie on one great circle and their lengths simply add: a circular slide rule inside SO(3).