  For two solutions of the WDVV equations that are related by two types of
symmetries of the equations given by Dubrovin, we show that the associated
principal hierarchies of integrable systems are related by certain reciprocal
transformation, and the tau functions of the hierarchies are either identical
or related by a Legendre transformation. We also consider relationships between
the Virasoro constraints and topological deformations of the principal
hierarchies.
