The distribution of horizontal subspaces is always Lagrangian not only for Riemannian metrics, but also for the Ehresmann connection associated to Finsler metrics and the Levi-Civita connection of pseudo-Riemannian metrics. It may not be Lagrangian for general sprays though. 

See my answer to this question https://mathoverflow.net/questions/127319/intuition-for-levi-civita-connection-via-hamiltonian-flows/127394#127394