We present a new paradigm for unification arising out of a
technique commonly used in cryptographic protocol analysis tools
that employ unification modulo equational theories.  This
paradigm relies on: (i) a decomposition of an equational theory
into (R,E) where R is confluent, terminating, and coherent modulo
E, and (ii) on reducing unification problems to a set of problems
$s =_{}^{?} t$ under the constraint that t remains
R/E-\emph{irreducible}.  We call this unification method
asymmetric unification because of the asymmetric irreducibility
constraint.  We first present the generic asymmetric unification,
and then outline an approach for converting conventional
unification algorithms to asymmetric ones, demonstrating it for
exclusive-or with uninterpreted function symbols.  We demonstrate
how asymmetric unification can improve the performance of
cryptographic protocol analysis tools by running the algorithm on
a set of benchmark problems.  We also give results on the
complexity and decidability of asymmetric unification.