Derived equivalences of K3 surfaces and orientation

Every Fourier--Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive directions. This leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical Torelli theorem for K3 surfaces determining all Hodge isometries that are induced by automorphisms.
The appendix contains a description of Lieblich's obstruction for the deformation of complexes in families in terms of Kodaira--Spencer and Atiyah classes which is of independent interest.