  The Bender-Knuth involutions on semistandard Young tableaux are known to
coincide with the tableau switching on horizontal border strips of two adjacent
letters, together with the swapping of those letters. Motivated by this
coincidence and using the shifted tableau switching due to Choi, Nam and Oh
(2019), we consider a shifted version of the Bender-Knuth involutions and
define a shifted version of the Berenstein-Kirillov group (1995). Similarly to
the classical case, the shifted version of the Berenstein-Kirillov group also
acts on the straight-shaped shifted tableau crystals introduced by Gillespie,
Levinson and Purbhoo (2020), via partial Sch\"utzenberger involutions, thus
coinciding with the action of the cactus group on the same crystal, due to the
author. Following the works of Halacheva (2016, 2020), and Chmutov, Glick and
Pylyavskyy (2020), on the relation between the actions of the
Berenstein-Kirillov group and the cactus group on a crystal of straight-shaped
Young tableaux, we also show that the shifted Berenstein-Kirillov group is
isomorphic to a quotient of the cactus group. Not all the known relations that
hold in the classic Berenstein-Kirillov group need to be satisfied by the
shifted Bender-Knuth involutions, but the ones implying the relations of the
cactus group are verified. Hence, we have an alternative presentation for the
cactus group in terms of the shifted Bender-Knuth involutions. We also use the
shifted growth diagrams due to Thomas and Yong (2016) to provide an alternative
proof concerning the mentioned cactus group action.
