  YbB$_6$ is a predicted topological insulator, with experimental evidence for
conducting surface states of yet-unproven origin. However, its lack of a
natural cleavage plane, and resultant surface-dependent polarity, has obscured
its study. We use scanning tunneling microscopy to image the cleaved surface of
YbB$_6$, exhibiting several coexisting terminations with distinct atomic
structures. Our spectroscopic measurements show band-bending between the
terminations, resulting in both conducting and fully-gapped regions. In the
conductive regions, we observe spectral peaks that are suggestive of van Hove
singularities arising from Rashba spin-split quantum well states. The
insulating regions rule out the possibility that YbB$_6$ is a strong
topological insulator, while the spin-polarized conducting regions suggest
possible utility for spintronic devices.
