  Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries
between them; that is, isomorphisms of their cohomologies which respect certain
natural lattice structures and Hodge structures. We prove a criterion for when
a given Hodge isometry arises in this way. In particular, we describe the image
of the representation which associates to any autoequivalence of a twisted K3
surface its realization in cohomology: this image is a subgroup of index one or
two in the group of all Hodge isometries of the twisted K3 surface. We show
that both indices can occur.
