  For certain pseudo-Anosov flows $\phi$ on closed $3$-manifolds, unpublished
work of Agol--Gu\'eritaud produces a veering triangulation $\tau$ on the
manifold $M$ obtained by deleting $\phi$'s singular orbits. We show that $\tau$
can be realized in $M$ so that its 2-skeleton is positively transverse to
$\phi$, and that the combinatorially defined flow graph $\Phi$ embedded in $M$
uniformly codes $\phi$'s orbits in a precise sense. Together with these facts
we use a modified version of the veering polynomial, previously introduced by
the authors, to compute the growth rates of $\phi$'s closed orbits after
cutting $M$ along certain transverse surfaces, thereby generalizing work of
McMullen in the fibered setting. These results are new even in the case where
the transverse surface represents a class in the boundary of a fibered cone of
$M$.
  Our work can be used to study the flow $\phi$ on the original closed
manifold. Applications include counting growth rates of closed orbits after
cutting along closed transverse surfaces, defining a continuous, convex entropy
function on the `positive' cone in $H^1$ of the cut-open manifold, and
answering a question of Leininger about the closure of the set of all stretch
factors arising as monodromies within a single fibered cone of a $3$-manifold.
This last application connects to the study of endperiodic automorphisms of
infinite-type surfaces and the growth rates of their periodic points.
