Lyα nebulae" are giant (∼100 kpc), glowing gas clouds in the distant universe. The origin of their extended Lyα emission remains a mystery. Some models posit that Lyα emission is produced when the cloud is photoionized by UV emission from embedded or nearby sources, while others suggest that the Lyα photons originate from an embedded galaxy or active galactic nucleus (AGN) and are then resonantly scattered by the cloud. At least in the latter scenario, the observed Lyα emission will be polarized. To test these possibilities, we are conducting imaging polarimetric observations of seven Lyα nebulae. Here we present our results for LABd05, a cloud at z=2.656 with an obscured, embedded AGN to the northeast of the peak of Lyα emission. We detect significant polarization. The highest polarization fractions P are ∼10%-20% at ∼20-40 kpc southeast of the Lyα peak, away from the AGN. The lowest P, including upper limits, are ∼5% and lie between the Lyα peak and AGN. In other words, the polarization map is lopsided, with P increasing from the Lyα peak to the southeast. The measured polarization angles θ are oriented northeast, roughly perpendicular to the P gradient. This unique polarization pattern suggests that (1) the spatially offset AGN is photoionizing nearby gas and (2) escaping Lyα photons are scattered by the nebula at larger radii and into our sightline, producing tangentially oriented, radially increasing polarization away from the photoionized region. Finally we conclude that the interplay between the gas density and ionization profiles produces the observed central peak in the Lyα emission. This also implies that the structure of LABd05 is more complex than assumed by current theoretical spherical or cylindrical models.