  We show that the symplectic reduction of the dynamics of $N$ point vortices
on the plane by the special Euclidean group $\mathsf{SE}(2)$ yields a
Lie--Poisson equation for relative configurations of the vortices.
Specifically, we combine symplectic reduction by stages with a dual pair
associated with the reduction by rotations to show that the
$\mathsf{SE}(2)$-reduced space with non-zero angular impulse is a coadjoint
orbit. This result complements some existing works by establishing a
relationship between the symplectic/Hamiltonian structures of the original and
reduced dynamics. We also find a family of Casimirs associated with the
Lie--Poisson structure including some apparently new ones. We demonstrate
through examples that one may exploit these Casimirs to show that some shape
dynamics are periodic.
