  The Casimir interaction between two objects, or between an object and a
plane, depends on their relative orientations. We make these angular
dependences explicit by considering prolate or oblate spheroids. The variation
with orientation is calculated exactly at asymptotically large distances for
the electromagnetic field, and at arbitrary separations for a scalar field. For
a spheroid in front of a mirror, the leading term is orientation independent,
and we find the optimal orientation from computations at higher order.
