Also, "double" rotations, which have no analogue in 3D space. The extra degree of freedom in 4D lets you perform 2 orthogonal rotations. This means that you can rotate around in 4D space, leaving only the origin fixed. In 3D rotations always fix some axis.

Trying to think about this stuff can be mind-bending, since it's natural to try to visualize things. So to give a little intuition, consider the coordinates of a 4D vector (x,y,z,w) and notice that we can spin around in both the x-y and z-w planes at the same time without interfering with each other!