  We apply the previous calculations of Chow-Witt rings of Grassmannians to
develop an oriented analogue of the classical Schubert calculus. As a result,
we get complete diagrammatic descriptions of the ring structure in Chow-Witt
rings and twisted Witt groups. In the resulting arithmetic refinements of
Schubert calculus, the multiplicity of a solution subspace is a quadratic form
encoding additional orientation information. We also discuss a couple of
applications, such as a Chow-Witt version of the signed count of balanced
subspaces of Feh\'er and Matszangosz.
