An algebraic vector bundle is oriented if its determinant is trivial, and is said to be metalinear (or quadratically oriented) if its determinant admits a square root. These bundles play an important role in motivic homotopy theory and quadratically enriched enumerative geometry, but their origins reach centuries earlier to the theory of theta-characteristics in the 1800's. Following contemporary work in motivic homotopy theory, we now know that metalinear bundles over a smooth affine variety over a field are classified by homotopy classes of maps into some universal classifying space, the cohomology of which is the home for characteristic classes of metalinear bundles. We compute the oriented Chow groups of this space, exploring the connections both with the theory of torsors and with classical topology. This is joint work with Matthias Wendt.