  We provide a geometric interpretation for the connecting homomorphism in the
localization sequence of Hermitian $K$-theory. As an application, we compute
the Hermitian $K$-theory of projective bundles and Grassmannians in the regular
case. We provide an explicit basis for Hermitian $K$-theory of Grassmannians,
which is indexed by even Young diagrams together with another special class of
Young diagrams, that we call $\textit{buffalo-check}$ Young diagrams. To
achieve this, we develop pushforwards and pullbacks in Hermitian $K$-theory
using Grothendieck's residue complexes, and we establish fundamental theorems
for those pushforwards and pullbacks, including base change, projection, and
excess intersection formulas.
