  We report on VLT/FORS2 imaging polarimetry observations in the $R_{\rm
special}$ band of WISE J011601.41-050504.0 (W0116-0505), a heavily obscured
hyper-luminous quasar at $z=3.173$ classified as a Hot, Dust-Obscured Galaxy
(Hot DOG) based on its mid-IR colors. Recently, Assef et al. (2020) identified
W0116-0505 as having excess rest-frame optical/UV emission, and concluded this
excess emission is most likely scattered light from the heavily obscured AGN.
We find that the broad-band rest-frame UV flux is strongly linearly polarized
(10.8$\pm$1.9\%, with a polarization angle of 74$\pm$9~deg), confirming this
conclusion. We analyze these observations in the context of a simple model
based on scattering either by free electrons or by optically thin dust,
assuming a classical dust torus with polar openings. Both can replicate the
degree of polarization and the luminosity of the scattered component for a
range of geometries and column densities, but we argue that optically thin dust
in the ISM is the more likely scenario. We also explore the possibility that
the scattering medium corresponds to an outflow recently identified for
W0116-0505. This is a feasible option if the outflow component is bi-conical
with most of the scattering occurring at the base of the receding outflow. In
this scenario the quasar would still be obscured even if viewed face on, but
might appear as a reddened type 1 quasar once the outflow has expanded. We
discuss a possible connection between blue-excess Hot DOGs, extremely red
quasars (ERQs), reddened type 1 quasars, and unreddened quasars that depends on
a combination of evolution and viewing geometry.
