  We undertake a systematic study of the infinitesimal geometry of the Thurston
metric, showing that the topology, convex geometry and metric geometry of the
tangent and cotangent spheres based at any marked hyperbolic surface
representing a point in Teichm{\"u}ller space can recover the marking and
geometry of this marked surface. We then translate the results concerning the
infinitesimal structures to global geometric statements for the Thurston
metric, most notably deriving rigidity statements for the Thurston metric
analogous to the celebrated Royden theorem.
