What is the physical meaning of the principal axes of inertia? I used to think that the axes of inertia are, in some sense, the only axes about which the body can rotate without the angular momentum "slipping" to other axes. In other words, I thought they are the only axes around which the body can have a motion of simple rotation about an axis, and any attempt to rotate the body around a non-principal axis wil result in a complex motion, consisting of a superposition of rotations around more than one principal axes (to put it differently, I thought principal axes are analogous to normal modes in vibrating systems, where the system can vibrate in a single frequency only if it's a a normal mode).
However, torque-free precession - or the general motion of a symmetric, non-spherical top - shows this is not the case. A symmetric, non-spherical top in general has a spin around its top axis, plus an additional spin around an axis that can make an angle with the top axis - and is, in general, non-principal. So what do the principal axes of inertia mean? What is their physical interpretation (since every book I read just says they are the eigenspaces of the inertia tensor, which is a statement that lacks any physical meaning)?