  Consider an arbitrary field $K$ and a finite-dimensional vector space $X$
over $K$ equipped with a, possibly degenerate, symplectic form $\omega$. Given
a spanning subset $S$ of $X$, for each $k$ in $K$ and each vector $s$ in $S$,
consider the symplectic transvection mapping a vector $x$ to $x+k\omega(x,s)s$.
The group generated by these transvections has been extensively studied, and
its orbit structure is known. In this paper, we obtain corresponding results
for the orbits of the dual action on $X^\ast$. As for the non-dual case, the
analysis gets harder when the field contains only two elements. For that field,
the dual transvection group is equivalent to a game known as the lit-only sigma
game, played on a graph. Our results provide a complete solution to the
reachability problem of that game, previously solved only for some special
cases.
