  On an oriented 4-manifold, we examine the geometry that arises when the
curvature operator of a Riemannian or Lorentzian metric $g$ commutes, not with
its own Hodge star operator, but rather with that of another semi-Riemannian
metric $h$ that is a suitable deformation of $g$. We classify the case when one
of these metrics is Riemannian and the other Lorentzian by generalizing the
concept of Petrov Type from general relativity; the case when $h$ is
split-signature is also examined. The "generalized Petrov Types" so obtained
are shown to relate to the critical points of $g$'s sectional curvature, and
sometimes yield unique normal forms. They also carry topological information
independent of the Hitchin-Thorpe inequality, and yield a direct geometric
formulation of "almost-Einsten" metric via the Ricci or sectional curvature of
$g$.
