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>~F5) hosting directly imaged planets and brown dwarf companions. As part of this effort, we have acquired new vsini* 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_-11_^+10^% of the sample) are likely misaligned, while the remaining 10 host stars are consistent with spin- orbit alignment. As an ensemble, the minimum obliquity distribution between 10 and 250au 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.