It is known that for a strong Dubreil-Jacotin semigroup the propeq of being perfect is equivalent to that of being naturally ordered. Here we consider the existence of a smallest idempotent and the analogous notion of being dually perfect, whch we show is not equivalent to being dually naturally ordered. The main objective of the paper is to obtain a suucture theorem for dually perfect orthodox Dubreil-Jacotinsemigroups on which Green's relations are regular.