  The canonical involution of a double (=iterated) tangent bundle may be
dualized in different ways to yield relations between the Tulczyjew
diffeomorphism, the Poisson anchor associated with the standard symplectic
structure on the cotangent space,and the reversal diffeomorphism. We show that
the constructions which yield these maps extend very generally to the double
Lie algebroids of double Lie groupoids, where they play a crucial role in the
relations between double Lie algebroids and Lie bialgebroids.
