A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed -forms. Another family of well-known examples are projectively flat Finsler metrics on the -sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let be a Riemannian–Finsler metric on a closed surface , and be a small antisymmetric potential on that is a natural generalization of -form (see Sec. 1). If the Randers-type Finsler metric is geodesically reversible, and the geodesic flow of is topologically transitive, then we prove that must be a closed -form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the -sphere of constant positive curvature.