  We establish the twisted crystallographic T-duality, which is an isomorphism
between Freed-Moore twisted equivariant K-groups of the position and momentum
tori associated to an extension of a crystallographic group. The proof is given
by identifying the map with the Dirac homomorphism in twisted
Chabert--Echterhoff KK-theory. We also illustrate how to exploit it in K-theory
computations.
