The "exceptional" isomorphism of Dynkin diagrams A_1^2 = D_2 classically gives rise to a bijection between quadric surfaces and quadratic Weil restrictions of Severi-Brauer curves. If S is a regular integral surface, D a smooth divisor, and T -> S a flat double cover branched along D, then I'll describe an analogous bijection between quadric surface bundles over S with degeneration along D and Azumaya quaternion algebras on T. A key ingredient is the study of orthogonal group schemes, which are smooth yet non-reductive, of quadratic bundles with "simple" degeneration along a smooth divisor. I'll present two applications: constructing counter-examples to the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms over function fields of surfaces over algebraically closed fields; and providing a new proof, employing results on moduli of twisted sheaves, of the global Torelli theorem for cubic fourfolds containing a plane. This is joint work with R. Parimala and V. Suresh.