For a subgroup W of the hyperoctahedral group O-n which is generated by reflections, we consider the linear dependence matroid M-W on the column vectors corresponding to the reflections in W. We determine all possible automorphism groups of M-W and determine when W congruent to Aut(M-W). This allows us to connect combinatorial and geometric symmetry. Applications to zonotopes are also considered. Signed graphs are used as a tool for constructing the automorphisms.