  Let $k$ be a regular ring, and let $A,B$ be essentially finite type
$k$-algebras. For any functor $F:{D}(A)\times\dots\times{D}(A)\to{D}(B)$
between their derived categories, we define its twist
$F^{!}:{D}(A)\times\dots\times{D}(A)\to{D}(B)$ with respect to dualizing
complexes, generalizing Grothendieck's construction of $f^{!}$. We show that
relations between functors are preserved between their twists, and deduce that
various relations hold between derived Hochschild (co)-homology and the $f^{!}$
functor. We also deduce that the set of isomorphism classes of dualizing
complexes over a ring (or a scheme) form a group with respect to derived
Hochschild cohomology, and that the twisted inverse image functor is a group
homomorphism.
