  All four circumbinary (CB) protoplanetary disks orbiting short-period ($P <
20$ day) double-lined spectroscopic binaries (SB2s)---a group that includes UZ
Tau E, for which we present new ALMA data---exhibit sky-plane inclinations
$i_{\rm disk}$ which match, to within a few degrees, the sky-plane inclinations
$i_\star$ of their stellar hosts. Although for these systems the true mutual
inclinations $\theta$ between disk and binary cannot be directly measured
because relative nodal angles are unknown, the near-coincidence of $i_{\rm
disk}$ and $i_\star$ suggests that $\theta$ is small for these most compact of
systems. We confirm this hypothesis using a hierarchical Bayesian analysis,
showing that 68% of CB disks around short-period SB2s have $\theta <
3.0^\circ$. Near co-planarity of CB disks implies near co-planarity of CB
planets discovered by Kepler, which in turn implies that the occurrence rate of
close-in CB planets is similar to that around single stars. By contrast, at
longer periods ranging from $30-10^5$ days (where the nodal degeneracy can be
broken via, e.g., binary astrometry), CB disks exhibit a wide range of mutual
inclinations, from co-planar to polar. Many of these long-period binaries are
eccentric, as their component stars are too far separated to be tidally
circularized. We discuss how theories of binary formation and disk-binary
gravitational interactions can accommodate all these observations.
