  We prove several conjectures about the cohomology of Deligne-Lusztig
varieties: invariance under conjugation in the braid group, behaviour with
respect to translation by the full twist, parity vanishing of the cohomology
for the variety associated with the full twist. In the case of split groups of
type $A$, and using previous results of the second author, this implies
Brou\'e-Michel's conjecture on the disjointness of the cohomology for the
variety associated to any good regular element. That conjecture was inspired by
Brou\'e's abelian defect group conjecture and the specific form Brou\'e
conjectured for finite groups of Lie type.
