  We prove that the action of the Grothendieck-Teichm\"uller group on the genus
completed properad of (homotopy) Lie bialgebras commutes with the reversing
directions involution of the latter. We also prove that every universal
quantization of Lie bialgebras is homotopy equivalent to the one which commutes
with the duality involution exchanging Lie bracket and Lie cobracket. The
proofs are based on a new result in the theory of oriented graph complexes
(which can be of independent interest) saying that the involution on an
oriented graph complex that changes all directions on edges induces the
identity map on its cohomology.
