  We introduce a new geometrical invariant of CR manifolds of hypersurface
type, which we dub the "Levi core" of the manifold. When the manifold is the
boundary of a smooth bounded pseudoconvex domain, we show how the Levi core is
related to two other important global invariants in several complex variables:
the Diederich--Forn{\ae}ss index and the D'Angelo class (namely the set of
D'Angelo forms of the boundary). We also show that the Levi core is trivial
whenever the domain is of finite-type in the sense of D'Angelo, or the set of
weakly pseudoconvex points is contained in a totally real submanifold, while it
is nontrivial if the boundary contains a local maximum set. As corollaries to
the theory developed here, we prove that for any smooth bounded pseudoconvex
domain with trivial Levi core the Diederich--Forn{\ae}ss index is one and the
$\overline{\partial}$-Neumann problem is exactly regular (via a result of Kohn
and its generalization by Harrington). Our work builds on and expands recent
results of Liu and Adachi--Yum.
