  We prove that there exists a geometric bijection between the sets of adjoint
and coadjoint orbits of a semidirect product, provided a similar bijection
holds for particular subgroups. We also show that under certain conditions the
homotopy types of any two orbits in bijection with each other are the same. We
apply our theory to the examples of the affine group and the Poincar\'{e}
group, and discuss the limitations and extent of this result to other groups.
