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Recently, Gillespie, Levinson and Purbhoo introduced a crystal-like structurefor shifted tableaux, called the shifted tableau crystal. We introduce ashifted analogue of the crystal reflection operators, which coincides with therestriction of the shifted Sch\"utzenberger involution to any primed intervalof two adjacent letters. Unlike type $A$ Young tableau crystals, theseoperators do not realize an action of the symmetric group on the shiftedtableau crystal because braid relations do not hold. We exhibit a naturalinternal action of the cactus group, realized by restrictions of the shiftedSch\"utzenberger involution on primed intervals of the underlying crystalalphabet.

An action of the cactus group on shifted tableau crystals