Construction of Cocycle Bicrossproducts by Twisting
  We consider a generalisation of the Majid's mirror product of a Hopf algebra H, when one of the components of the product is replaced by a twist. This leads to a new "twisted mirror product" construction for cocycle bicrossproduct Hopf algebra where c is a 2-cocycle of "twisting" type. As an application, we obtain a canonical cocycle bicrossproduct for any Hopf algebra H. 
G. Bogdanoff, I. Bogdanoff