  It was recently shown that the string theory duals of certain deformations of
the N=4 gauge theory can be obtained by a combination of T-duality
transformations and coordinate shifts. Here we work out the corresponding
procedure of twisting the dual integrable spin chain and its Bethe ansatz. We
derive the Bethe equations for the complete twisted N=4 gauge theory at one and
higher loops. These have a natural generalization which we identify as twists
involving the Cartan generators of the conformal algebra. The underlying model
appears to be a form of noncommutative deformation of N=4 SYM.
