In four spacetime dimensions gravitational plane waves (a special case of the plane-fronted waves with parallel rays) admit a 5 parameter isometry group. We generalize this group to n-dimensions and explore some special features of spacetimes admitting this isometry group. In particular, it is shown that every generally covariant rank-2 symmetric tensor field constructed from a metric with plane wave symmetry will vanish except multiples of the metric and Ricci tensors. We show that, in four spacetime dimensions, a particular enlargement of the plane wave symmetry group is enough to force the group-invariant metrics to satisfy all generally covariant vacuum equations. We comment on the construction of a “mini-superspace” description of spacetimes admitting plane wave symmetry.