Moments and Product of Inertia

Now,

But the magnitude of the position vector from the origin to a particle is independent of the inclination of the reference at the origin. Thus, the sum of the moments of inertia at a point in space for a given body clearly is an invariant with respect to roation of axes. This is called first invariant.

It can also be easily seen that if two axes form a plane of symmetry for the mass distribution of a body, the product of inertia having as an index the coordinate that is normal to the plane of symmetry will be zero.