  We study the plane-symmetric collision of two gravitational waves and
describe the global spacetime geometry generated by this collision. To this
end, we formulate the characteristic initial value problem for the Einstein
equations, when Goursat data describing the incoming waves are prescribed on
two null hypersurfaces. We then construct a global solution representing a
cyclic spacetime based on junction conditions associated with a prescribed
singularity scattering map (a notion recently introduced by the authors). From
a mathematical analysis standpoint, this amounts to a detailed analysis of the
Goursat and Fuchsian initial value problems associated with singular hyperbolic
equations, when junction conditions at interfaces are prescribed. In our
construction, we introduce a partition into monotonicity diamonds (that is,
suitable spacetime domains) and we construct the solution by concatenating
domains across interfaces of timelike, null, or spacelike type.
