  Let $\varphi$ be a transitive pseudo-Anosov flow on an oriented, compact
$3$-manifold $M$, possibly with toral boundary. We characterize the surfaces in
$M$ that are (almost) transverse to $\phi$. When $\varphi$ has no perfect fits
(e.g. $\varphi$ is the suspension flow of a pseudo-Anosov homeomorphism), we
prove that any Thurston-norm minimizing surface $S$ that pairs nonnegatively
with the closed orbits of $\varphi$ is almost transverse to $\varphi$, up to
isotopy. This answers a question of Cooper--Long--Reid. Our main tool is a
correspondence between surfaces that are almost transverse to $\varphi$ and
those that are relatively carried by any associated veering triangulation. The
correspondence also allows us to investigate the uniqueness of almost
transverse position, to extend Mosher's Transverse Surface Theorem to the case
with boundary, and more generally to characterize when relative homology
classes represent Birkhoff surfaces.
