A quasi-Coxeter category is a braided tensor category which carries an action of a generalised braid group B_W on the tensor powers of its objects. The data which defines the action of B_W is similar in flavour to the associativity constraints in a monoidal category, but is related to the coherence of a family of fiber functors on C. I will outline how to construct such a structure on integrable, category O representations of a symmetrisable Kac-Moody algebra g, in a way that incorporates the monodromy of the KZ and Casimir connections of g. The rigidity of this structure implies in particular that the monodromy of the latter connection is given by the quantum Weyl group operators of the quantum group U_h(g). This is joint work with Andrea Appel.