  We construct semi-infinite billiard domains which reverse the direction of
most incoming particles. We prove that almost all particles will leave the open
billiard domain after a finite number of reflections. Moreover, with high
probability the exit velocity is exactly opposite to the entrance velocity, and
the particle's exit point is arbitrarily close to its initial position. The
method is based on asymptotic analysis of statistics of entrance times to a
small interval for irrational circle rotations. The rescaled entrance times
have a limiting distribution in a limit when the number of iterates tends to
infinity and the length of the interval vanishes. The proof of the main results
follows from the study of related limiting distributions and their regularity
properties.
