  We illustrate the general point of view developed in [SIAM J. Math. Anal.,
51(6), 4356-4381] that can be described as a variation of Helgason's theory of
dual $G$-homogeneous pairs $(X,\Xi)$ and which allows us to prove intertwining
properties and inversion formulae of many existing Radon transforms. Here we
analyze in detail one of the important aspects in the theory of dual pairs,
namely the injectivity of the map label-to-manifold $\xi\to\hat\xi$ and we
prove that it is a necessary condition for the irreducibility of the
quasi-regular representation of $G$ on $L^2(\Xi)$. We further explain how the
theory in [SIAM J. Math. Anal., 51(6), 4356-4381] applies to the classical
Radon and X-ray transforms in $\mathbb R^3$.
