  In this paper we discuss the concept of cosymmetries and co--recursion
operators for difference equations and present a co--recursion operator for the
Viallet equation. We also discover a new type of factorisation for the
recursion operators of difference equations. This factorisation enables us to
give an elegant proof that the recursion operator given in arXiv:1004.5346 is
indeed a recursion operator for the Viallet equation. Moreover, we show that
this operator is Nijenhuis and thus generates infinitely many commuting local
symmetries. This recursion operator and its factorisation into Hamiltonian and
symplectic operators can be applied to Yamilov's discretisation of the
Krichever-Novikov equation.
