  A new approach is given to property $(P_q)$ defined by Catlin for $q=1$ in a
global and by Sibony in a local context, subsequently extended by Fu-Straube
for $q>1$. This property is known to imply compactness and global regularity in
the $\bar\partial$-Neumann problem by a result of Kohn-Nirenberg, as well as
condition $R$ by a result of Bell-Ligocka. In particular, we provide a
self-contained proof of property $(P_q)$ for pseudoconvex hypersurfaces of
finite D'Angelo $q$-type, the case originally studied by Catlin. Moreover, our
proof covers more general classes of hypersurfaces inspired by a recent work of
Huang-Yin. Proofs are broken down into isolated steps, some of which do not
require pseudoconvexity.
  Our tools include: a new multitype invariant based on distinguished nested
sequences of $(1,0)$ subbundles, defined in terms of derivatives of the Levi
form; real and complex formal orbits; $k$-jets of functions relative to pairs
of formal submanifolds; relative contact orders generalizing the usual contact
orders; a new notion of supertangent vector fields having higher than expected
relative contact orders; and a formal variant of a result by Diederich-Forn\ae
ss arising as a key step in their proof of Kohn's ideal termination in the
real-analytic case.
