All maximally entangling unitary operators

We characterize all maximally entangling bipartite unitary operators, acting on systems $A$ and $B$ of arbitrary finite dimensions ${d}_{A}\ensuremath{\le}{d}_{\phantom{\rule{-0.16em}{0ex}}B}$, when ancillary systems are available to both parties. Several useful and interesting consequences of this characterization are discussed, including an understanding of why the entangling and disentangling capacities of a given (maximally entangling) unitary can differ and a proof that these capacities must be equal when ${d}_{A}={d}_{B}$.