  We present a study of resonant inelastic X-ray scattering (RIXS) spectra
collected at the rare-earth $L$ edges of divalent hexaborides YbB$_6$ and
EuB$_6$. In both systems, RIXS-active features are observed at two distinct
resonances separated by $\sim10$ eV in incident energy, with angle-dependence
suggestive of distinct photon scattering processes. RIXS spectra collected at
the divalent absorption peak strongly resemble the unoccupied 5$d$ density of
states calculated using density functional theory, an occurrence we ascribe to
transitions between weakly-dispersing 4$f$ and strongly dispersing 5$d$ states.
In addition, anomalous resonant scattering is observed at higher incident
energy, where no corresponding absorption feature is present. Our results
suggest the far-reaching utility of $L$-edge RIXS in determining the
itinerant-state properties of $f$-electron materials.
