Noncommutative spacetime symmetries: twist versus covariance

We prove that the Moyal product is covariant under linear affine spacetime transformations. From the covariance law, by introducing an (x, Theta)-space where the spacetime coordinates and the noncommutativity matrix components are on the same footing, we obtain a noncommutative representation of the affine algebra, its generators being differential operators in (x, Theta)-space. As a particular case, theWeyl Lie algebra is studied and known results for Weyl invariant noncommutative field theories are rederived in a nutshell. We also show that this covariance cannot be extended to spacetime transformations generated by differential operators whose coefficients are polynomials of order larger than 1.We compare our approach with the twist-deformed enveloping algebra description of spacetime transformations.