A Finsler space (M,Σ) is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason [13], it is shown that a geodesically reversible Finsler metric of constant flag curvature on the...