We discuss recent results regarding $C^\infty$ -isomorphism properties of weighted normal operators of the X-ray transform on manifolds with boundary, in joint work with Francois Monard and Rohit Kumar Mishra. The crux of the result depends on understanding the Singular Value Decomposition of weighted X-ray transforms/backprojection operators, which itself can be obtained via intertwining with certain degenerately elliptic differential operators. We also discuss recent work with Francois Monard on developing tools to study such degenerately elliptic operators even further. Such tools include a scale of Sobolev spaces which take into account behavior up to the boundary, as well as generalizations of Dirichlet and Neumann traces called boundary triplets associated to degenerately elliptic operators which pick out the first and second most singular terms of a function near the boundary.