= Degeneracy of principal moments of inertia
If two <principal moments of inertia> coincide, for example $I_1=I_2\ne I_3$, every direction in their two-dimensional <eigenspace> is a <principal axis>. The perpendicular axis is determined, but there is no privileged orthogonal pair within that plane. Rotating the pair in the plane leaves the diagonal <inertia tensor> unchanged. If all three <principal moments of inertia> coincide, $\mathbf I_O=I_0\mathbf1$ and every axis through $O$ is principal. This follows from the <eigenvalue multiplicity> and <eigenspace>, rather than from a failure of <diagonalization>. <Newtonian Dynamics by Richard Fitzpatrick/Matrix Eigenvalue Theory>{ref}
Equal <principal moments of inertia> do not prove that a body's outline has continuous rotational symmetry. For example, six equal <point masses> at $(\pm a,0,0)$, $(0,\pm a,0)$ and $(0,0,\pm a)$ have $\mathbf I_O=4ma^2\mathbf1$. This discrete layout has the same axial <moment of inertia> in every direction, although its actual <mass> distribution changes under most rotations. Conversely, even when all three <principal moments of inertia> are distinct, <principal axes> are unoriented lines: reversing an <eigenvector>'s sign gives the same axis. Near repeated <eigenvalues>, small measurement errors can rotate fitted <principal axes> substantially within the nearly degenerate <eigenspace>, so quoted directions need an uncertainty assessment as well as quoted <principal moments of inertia>.
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