The logo is a projection into our regular 3D space of a surface that exists in the 3D space of rotations, SO(3). Some sort of flat projection is necessary because SO(3) is a curved space locally like a hypersphere - the 3D surface swept out about a point in 4D space by a fixed length radius. Hyperspheres are hard to get one's head around without a suitable map, which is where this surface comes in.

The surface shows how a sequence of two perpendicular rotations unfold in rotation space. Physically, it represents the constraint that the two rotation axes are perpendicular, a constraint that is realised in any mechanism consisting of a pair of revolute joints (hinges) with their axes at right angles - a steered wheel or a universal joint, for example.

Topologically it is a torus, usually seen as a doughnut-shaped surface in regular space but possessing much more symmetry in SO(3) where the inside and the outside of the torus are identically shaped spaces. The torus contains a circle's worth of circles. Each line shown on the surface is a great-circle or geodesic, the equivalent of a straight line on SO(3).

Here is a 3D viewer allowing interactive rotation (300K download, requires a java-enabled browser).

I am working on a '4D viewer' which will allow interactive exploration of 4D rotations of the hypersphere.