  Given a Hopf algebra $A$ graded by a discrete group together with an action
of the same group preserving the grading, we define a new Hopf algebra, which
we call the graded twisting of $A$. If the action is adjoint, this new Hopf
algebra is a twist of $A$ by a pseudo-$2$-cocycle. Analogous construction can
be carried out for monoidal categories. As examples we consider graded
twistings of the Hopf algebras of nondegenerate bilinear forms, their free
products, hyperoctahedral quantum groups and $q$-deformations of compact
semisimple Lie groups. As applications, we show that the analogues of the
Kazhdan-Wenzl categories in the general semisimple case cannot be always
realized as representation categories of compact quantum groups, and for
genuine compact groups, we analyze quantum subgroups of the new twisted compact
quantum groups, providing a full description when the twisting group is cyclic
of prime order.
