Geometry of the commutator of two rotations

Question

Let $A$ and $B$ be two special orthogonal matrices, i.e. two rotations in some vector space. It is easy to imagine a rotation, at least in 3 dimensions. But I don't know how to imagine the commutator of two rotations, $ABA^TB^T$. What does this operator do to a vector, exactly? Of course it is also a rotation, but how is that rotation related to the ones performed by $A$ and $B$? Or, in other words, why should I care about the commutator? What is the geometry of it?