  We establish a one-to-one correspondence between Finsler structures on the
$2$-sphere with constant curvature $1$ and all geodesics closed on the one
hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci
curvature is positive definite and all of whose geodesics are closed on the
other hand. As an application of our duality result, we show that suitable
holomorphic deformations of the Veronese embedding $\mathbb{CP}(a_1,a_2)\to
\mathbb{CP}(a_1,(a_1+a_2)/2,a_2)$ of weighted projective spaces provide
examples of Finsler $2$-spheres of constant curvature whose geodesics are all
closed.
