  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.
