Can a Zhang twist be a cocycle twist

We provide sufficient conditions for a Zhang twist of a $\mathbb Z$-graded Hopf algebra $H$ to be again a Hopf algebra, to be an $H$-cleft object, or a 2-cocycle twist of $H$. In particular, we introduce the notion of a twisting pair for $H$ such that the Zhang twist of $H$ by such a pair is a 2-cocycle twist. This new notion is investigated in the context of various examples of Hopf algebras including Manin's universal quantum groups and quantized coordinate rings.