More examples of invariance under twisting

The so-called “invariance under twisting” for twisted tensor products of algebras is a result stating that, if we start with a twisted tensor product, under certain circumstances we can “deform” the twisting map and we obtain a new twisted tensor product, isomorphic to the given one. It was proved before that a number of independent and previously unrelated results from Hopf algebra theory are particular cases of this theorem. In this article we show that some more results from literature are particular cases of invariance under twisting, for instance a result of Beattie-Chen-Zhang that implies the Blattner-Montgomery duality theorem.