I was always puzzled by this fact.
A uniform cube has a sphere-shaped inertia ellipsoid. The sphere has a higher symmetry then the cube.
Is there any deep reason or implication behind it?
The inertia ellipsoid is computed from an integral about an axis - in other words you rotate the object. This will "smooth out" any symmetries and typically increase the symmetry.
Sorry this is a "early morning" intuitive explanation - maybe someone else will give you a more formal answer.