  The obliquity of a star, or the angle between its spin axis and the average
orbit normal of its companion planets, provides a unique constraint on that
system's evolutionary history. Unlike the Solar System, where the Sun's equator
is nearly aligned with its companion planets, many hot Jupiter systems have
been discovered with large spin-orbit misalignments, hosting planets on polar
or retrograde orbits. We demonstrate that, in contrast to stars harboring hot
Jupiters on circular orbits, those with eccentric companions follow no
population-wide obliquity trend with stellar temperature. This finding can be
naturally explained through a combination of high-eccentricity migration and
tidal damping. Furthermore, we show that the joint obliquity and eccentricity
distributions observed today are consistent with the outcomes of
high-eccentricity migration, with no strict requirement to invoke the other hot
Jupiter formation mechanisms of disk migration or in-situ formation. At a
population-wide level, high-eccentricity migration can consistently shape the
dynamical evolution of hot Jupiter systems.
