You're more likely to get an answer in the Programming/Customization forum.
There are SNAP and NX/Open functions that calculate moments of inertia and principal axes. And, if an object has a plane of symmetry, then the normal of that plane is a principal axis.
You could experiment with the interactive mass properties function, and confirm that knowledge of the principal axes helps with your problem.
Then, if you want to automate, post a question on the Programming/Customization forum.