  Recently, Gillespie, Levinson and Purbhoo introduced a crystal-like structure
for shifted tableaux, called the shifted tableau crystal. We introduce, on this
structure, a shifted version of the crystal reflection operators, which
coincide with the restrictions of the shifted Sch\"utzenberger involution to
any primed interval of two adjacent letters. Unlike type $A$ Young tableau
crystals, these operators do not realize an action of the symmetric group on
the shifted tableau crystal since the braid relations do not need to hold.
Following a similar approach as Halacheva, we exhibit a natural internal action
of the cactus group on this crystal, realized by the restrictions of the
shifted Sch\"utzenberger involution to all primed intervals of the underlying
crystal alphabet, containing, in particular, the aforesaid action of the
shifted crystal reflection operator analogues.
