  In recent work, Launois and Lenagan have shown how to construct a cocycle
twisting of the quantum Grassmannian and an isomorphism of the twisted and
untwisted algebras that sends a given quantum minor to the minor whose index
set is permuted according to the $n$-cycle $c=(1, 2, ..., n)$, up to a power of
$q$. This twisting is needed because $c$ does not naturally induce an
automorphism of the quantum Grassmannian, as it does classically and
semi-classically. We extend this construction to give a quantum analogue of the
action on the Grassmannian of the dihedral subgroup of $S_{n}$ generated by $c$
and $w_{0}$, the longest element, and this analogue takes the form of a
groupoid. We show that there is an induced action of this subgroup on the
torus-invariant prime ideals of the quantum Grassmannian and also show that
this subgroup acts on the totally nonnegative and totally positive
Grassmannians. Then we see that this dihedral subgroup action exists
classically, semi-classically (by Poisson automorphisms and anti-automorphisms,
a result of Yakimov) and in the quantum and nonnegative settings.
