We know that for a body with an axis of symmetry of order greater than 2 has two equal moments of inertia. But can the converse be proved? Namely, given only that two principal moments are equal, can it be deduced that an axis of symmetry of order greater than 2 exists for the body?
If the claim is false, is there a counterexample which happens to have two equal moments, but does not "look" symmetric?
The claim seems to be true to me, but beyond the observation that choice of two of the principal axes are arbitrary in a plane (due to degeneracy of the moment of inertia tensor), I do not know how to rigorously prove the existence of an axis of symmetry.
Are there standard ways to make (and prove) strong statements of symmetry in physics given a few calculated quantities?