  We characterize all maximally entangling bipartite unitary operators, acting
on systems $A,B$ of arbitrary finite dimensions $d_A\le d_B$, when use of
ancillary systems by both parties is allowed. 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 $d_A=d_B$.
