  Perverse-Hodge complexes are objects in the derived category of coherent
sheaves obtained from Hodge modules associated with Saito's decomposition
theorem. We study perverse-Hodge complexes for Lagrangian fibrations and
propose a symmetry between them. This conjectural symmetry categorifies the
"Perverse = Hodge" identity of the authors and specializes to Matsushita's
theorem on the higher direct images of the structure sheaf. We verify our
conjecture in several cases by making connections with variations of Hodge
structures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras.
