A formalism for the analysis of detailed photofragmentation experiments involving oriented/aligned parent molecules is presented. It is demonstrated for the photofragmentation of an oriented/aligned parent molecule that both the magnitudes and the relative-phases of the transition dipole matrix elements can be determined from angle integrated cross sections or the product orientation/alignment moments derived from them. Formulas are presented for extracting transition dipole matrix elements from the photodissociation of an aligned diatomic molecule to form aligned and/or oriented atomic products using linearly or circularly polarized photolysis lasers. While these detailed measurements are ideally performed using inhomogeneous field state-selected parent molecules, they can be also readily carried out by doing double-resonance photodissociation experiments with polarized lasers. The analysis presented here can, therefore, be used to extract the transition dipole matrix elements from these double-resonance photofragmentation experiments as well.