The inversion symmetry of the WDVV equations and tau functions

For two solutions of the WDVV equations that are related by the inversion symmetry, we show that the associated principal hierarchies of integrable systems are related by a reciprocal transformation, and the tau functions of the hierarchies are related by a Legendre-type transformation. We also consider relationships between the Virasoro constraints and the topological deformations of the principal hierarchies.