Another way of showing 4D rotations is to sweep out a surface. Just as a point can trace a line as it moves, we can create a surface from the path traced out by a line rotating in 4D. A huge range of different surfaces can be generated in this way, depending on the lines chosen, the projection and the axes of rotation. The following are just a few examples of such surfaces, all using a circle as the initial curve (it is interesting to note that if you start with a circle, it remains a circle throughout the rotation, this is because stereographic projection is conformal) :
The two videos above show the non-orientable surface known as a Klein bottle. It might not look like the usual representation of a Klein bottle but it is topologically the same surface, and I think this embedding of it (known as a Lawson klein bottle) is actually much more elegant.
An unusual embedding of the Mobius strip can also be generated in this way:
This is the a mobius band with a circular edge, also known as a Sudanese mobius band (I have no idea where the name comes from).