  We introduce intertwining operators among twisted modules or twisted
intertwining operators associated to not-necessarily-commuting automorphisms of
a vertex operator algebra. Let $V$ be a vertex operator algebra and let
$g_{1}$, $g_{2}$ and $g_{3}$ be automorphisms of $V$. We prove that for
$g_{1}$-, $g_{2}$- and $g_{3}$-twisted $V$-modules $W_{1}$, $W_{2}$ and
$W_{3}$, respectively, such that the vertex operator map for $W_{3}$ is
injective, if there exists a twisted intertwining operator of type
${W_{3}\choose W_{1}W_{2}}$ such that the images of its component operators
span $W_{3}$, then $g_{3}=g_{1}g_{2}$. We also construct what we call the
skew-symmetry and contragredient isomorphisms between spaces of twisted
intertwining operators among twisted modules of suitable types. The proofs of
these results involve careful analysis of the analytic extensions corresponding
to the actions of the not-necessarily-commuting automorphisms of the vertex
operator algebra.
