  The orientation between a star's spin axis and a planet's orbital plane
provides valuable information about the system's formation and dynamical
history. For non-transiting planets at wide separations, true stellar
obliquities are challenging to measure, but lower limits on spin-orbit
orientations can be determined from the difference between the inclination of
the star's rotational axis and the companion's orbital plane ($\Delta i$). We
present results of a uniform analysis of rotation periods, stellar
inclinations, and obliquities of cool stars (SpT $\gtrsim$ F5) hosting directly
imaged planets and brown dwarf companions. As part of this effort, we have
acquired new $v \sin i_*$ values for 22 host stars with the high-resolution
Tull spectrograph at the Harlan J. Smith telescope. Altogether our sample
contains 62 host stars with rotation periods, most of which are newly measured
using light curves from the Transiting Exoplanet Survey Satellite. Among these,
53 stars have inclinations determined from projected rotational and equatorial
velocities, and 21 stars predominantly hosting brown dwarfs have constraints on
$\Delta i$. Eleven of these (52$^{+10}_{-11}$% of the sample) are likely
misaligned, while the remaining ten host stars are consistent with spin-orbit
alignment. As an ensemble, the minimum obliquity distribution between 10-250 AU
is more consistent with a mixture of isotropic and aligned systems than either
extreme scenario alone--pointing to direct cloud collapse, formation within
disks bearing primordial alignments and misalignments, or architectures
processed by dynamical evolution. This contrasts with stars hosting directly
imaged planets, which show a preference for low obliquities. These results
reinforce an emerging distinction between the orbits of long-period brown
dwarfs and giant planets in terms of their stellar obliquities and orbital
eccentricities.
