Help Me! Question About Tensors and Projections

I need some help. I am working hard to find ways to bring more geometry into introductory calculus-based physics (and conceptual physics as well). By geometry, I mean specifically the geometry associated with vectors and tensors, and the information encoded therein. Yes, I said tensors.

I have been heavily influenced by these notes by Kip Thorne and Roger Blandford that form the basis of their forthcoming textbook. In particular, chapter 1 has deeply affected my view of what introductory physics could, and perhaps should, be like when an emphasis on geometry is provided. The discussion of tensors in these notes presents them as machines that take vectors as inputs and give a real number as output. This is the same approach taken in the famous general relativity textbook coauthored by Thorne (aka MTW) and in a fantastic new book by Jeevanjee. I won’t go into details here because I don’t think it’s necessary for my questions, but I look forward to exploring the approach in more depth in future posts.

Geometric entities, like vectors, have an existence independent of any coordinate system. For example, I can state that an object or particle has 35 units of momentum in a particular direction represented by an arrow and I can do so without choosing a coordinate system. I can project this momentum onto any arbitrary basis and resolve it into components if I want to. Fine. No problem.

But what about a quantity represented by a second rank tensor? What about moment of inertia (MoI0? MoI is a geometric quantity that has an existence independent of any coordinate system, but it isn’t a vector; it’s a second rank tensor that can be represented by a symmetric matrix.

So here is my question. How can I specificy a MoI in a coordinate-independent way analogous to doing so for a vector? For a vector, I can specificy a magnitude and draw a direction. What must I specify for MoI? I think I know the answer. I think I must specify the eigenvalues of the matrix representation of the MoI tensor and the principal axes to which these eigenvalues apply. These can then be projected onto a coordinate basis. Is that correct? If not, could you tell me if what I’m asking is even possible? I don’t see how it can’t be.