  We consider the construction of twisted tensor products in the category of
C*-algebras equipped with orthogonal filtrations and under certain assumptions
on the form of the twist compute the corresponding quantum symmetry group,
which turns out to be the generalised Drinfeld double of the quantum symmetry
groups of the original filtrations. We show how these results apply to a wide
class of crossed products of C*-algebras by actions of discrete groups. We also
discuss an example where the hypothesis of our main theorem is not satisfied
and the quantum symmetry group is not a generalised Drinfeld double.
