  We point out a symmetry exhibited by pairs of entangled states and discuss
its possible applications in quantum information. More specifically, we
consider quadripartite systems prepared in bipartite product states of the form
$|\Psi> = |\psi>_{12} \otimes |\psi>_{34}$ and let the uncorrelated subsystems
14 and 23 interact by a given unitary $U_{14}\otimes U_{23}$: we show that the
entanglement between the noninteracting subsystems 12 and 34 may induce
transparency, i.e. makes $|\Psi>$ an eigenstate of the unitary. We investigate
the occurrence of this phenomenon both in continuous variable and qubit
systems, and discuss its possible applications to bath engineering, double
swapping and remote inversion.
