Regularity of Hamilton-Jacobi equations when forward is backward

We introduce a general principle determining in certain cases the regularit y of the viscosity solutions of Hamilton-Jacobi equations. This principle s ays that if one can solve the equation forward in time from some initial da ta and then backward in time resulting in the same initial data, then the s olution must be C-1. Some cases are given when this holds as well as an exa mple when it does not. Convexity of either the hamiltonian or the initial d ata plays a crucial role throughout.