  It is well-known that granular mixtures that differ in size or shape
segregate when sheared. In the past, two mechanisms have been proposed to
describe this effect, and it is unclear if both exist. To settle this question,
we consider a bidisperse mixture of spheroids of equal volume in a rotating
drum, where the two mechanisms are predicted to act in opposite directions. We
present the first evidence that there are two \emph{distinct} segregation
mechanisms driven by relative \emph{over-stress}. Additionally, we showed that
for non-spherical particles, these two mechanisms can act in different
directions leading to a competition between the effects of the two. As a
result, the segregation intensity varies non-monotonically as a function of
$AR$, and at specific points, the segregation direction changes for both
prolate and oblate spheroids, explaining the surprising segregation reversal
previously reported. Consistent with previous results, we found that the
kinetic mechanism is dominant for (almost) spherical particles. Furthermore,
for moderate aspect ratios, the kinetic mechanism is responsible for the
spherical particles segregation to the periphery of the drum, and the gravity
mechanism plays only a minor role. Whereas, at the extreme values of $AR$, the
gravity mechanism notably increases and overtakes its kinetic counterpart.
