  Calm\`es and Fasel have shown that the twisted Witt groups of split flag
varieties vanish in a large number of cases. For flag varieties over
algebraically closed fields, we sharpen their result to an if-and-only-if
statement. In particular, we show that the twisted Witt groups vanish in many
previously unknown cases. In the non-zero cases, we find that the twisted total
Witt group forms a free module of rank one over the untwisted total Witt group,
up to a difference in grading.
  Our proof relies on an identification of the Witt groups of flag varieties
with the Tate cohomology groups of their K-groups, whereby the verification of
all assertions is eventually reduced to the computation of the (twisted) Tate
cohomology of the representation ring of a parabolic subgroup.
