  We construct the twist automorphism of open Richardson varieties inside the
flag variety of a complex semisimple algebraic group. We show that the twist
map preserves totally positive parts, and prove a Chamber Ansatz formula for
it. Our twist map generalizes the twist maps previously constructed by
Berenstein-Fomin-Zelevinsky, Marsh-Scott, and Muller-Speyer. We use it to
explain the relationship between the two conjectural cluster structures for
Richardson varieties studied by Leclerc and by Ingermanson.
