  We classify twistings of Grothendieck's differential operators on a smooth
variety $X$ in prime characteristic $p$. We prove isomorphism classes of
twistings are in bijection with $H^2(X,\mathbb{Z}_p(1))$, the degree 2, weight
1 syntomic cohomology of $X$. We also discuss the relationship between
twistings of crystalline and Grothendieck differential operators. Twistings in
mixed characteristic are also analyzed.
