  We present a novel analysis of a Radon transform, $R$, which maps an $L^2$
function of compact support to its integrals over smooth surfaces of revolution
with centers on an embedded hypersurface in $\mathbb{R}^n$. Using microlocal
analysis, we derive necessary and sufficient conditions relating to $R$ for the
Bolker condition to hold, which has implications regarding the existence and
location of image artifacts. We present a general inversion framework based on
Volterra equation theory and known results on the spherical Radon transform,
and we prove injectivity results for $R$. Several example applications of our
theory are discussed in the context of, e.g., Compton Scatter Tomography (CST)
and Ultrasound Reflection Tomography (URT). In addition, using the proposed
inversion framework, we validate our microlocal theory via simulation, and
present simulated image reconstructions of image phantoms with added noise.
