  A Finsler metric is geodesically reversible if geodesics remain geodesics
after a change of orientation. Asymmetric norms on vector spaces and Funk
metrics in the interior of convex bodies are examples of geodesically
reversible metrics that are not necessarily sums of reversible metrics and
closed 1-forms. However, there seem to be few such examples in closed
manifolds. In this paper the theory of volumes and areas on Finsler spaces is
applied to establish a number of rigidity theorems which partially explain this
paucity of examples. These rigidity results settle some hitherto unsolved cases
of Hilbert's fourth problem for asymmetric metrics.
