  Strong-field gravitational plane waves are often represented in either the
Rosen or Brinkmann forms. While these two metric ansatze are related by a
coordinate transformation, so that they should describe essentially the same
physics, they rather puzzlingly seem to treat polarization states quite
differently. Both ansatze deal equally well with + and X linear polarizations,
but there is a qualitative difference in they way they deal with circular,
elliptic, and more general polarization states. In this article we will develop
a general formalism for dealing with arbitrary polarization states in the Rosen
form of the gravitational wave metric, representing an arbitrary polarization
by a trajectory in a suitably defined two dimensional hyperbolic plane.
