  We study the classical and quantum mechanics of a three-dimensional stadium
billiard. It consists of two quarter cylinders that are rotated with respect to
each other by 90 degrees, and it is classically chaotic. The billiard exhibits
only a few families of nongeneric periodic orbits. We introduce an analytic
method for their treatment. The length spectrum can be understood in terms of
the nongeneric and unstable periodic orbits. For unequal radii of the quarter
cylinders the level statistics agree well with predictions from random matrix
theory. For equal radii the billiard exhibits an additional symmetry. We
investigated the effects of symmetry breaking on spectral properties. Moreover,
for equal radii, we observe a small deviation of the level statistics from
random matrix theory. This led to the discovery of stable and marginally stable
orbits, which are absent for un equal radii.
