  We prove a new systolic volume lower bound for non-orientable n-manifolds,
involving the stable 1-systole and the codimension 1 systole with coefficients
in Z_2. As an application, we prove that Lusternik-Schnirelmann category and
systolic category agree for non-orientable closed manifolds of dimension 3,
extending our earlier result in the orientable case. Finally, we prove the
homotopy invariance of systolic category.
