  A Coxeter system is called two-dimensional if its associated Davis complex is
two-dimensional (equivalently, every spherical subgroup has rank less than or
equal to 2). We prove that given a two-dimensional system (W,S) and any other
system (W,S') which yields the same reflections, the diagrams corresponding to
these systems are isomorphic, up to the operation of diagram twisting defined
by N. Brady, J. McCammond, B. Muhlherr, and W. Neumann. As a step in the proof
of this result, certain two-dimensional groups are shown to be reflection rigid
in the sense defined by the above authors, and a result concerning the strong
rigidity of two-dimensional systems is given in the final section.
