The dihedral group D, is, by definition, the (non-Abelian) group of symmetries of the n-sided regular polygon. It is well-known that the group of 12 transpositions and 12 inversions acting on the 12 pitch classes (T/I) is isomorphic to D12, as is the RiemannKlumpenhouwer Schritt/Wechsel group (S/W). It follows that T/I is isomorphic to S/W. However, in Lewin's terms, these two groups are "anti-isomorphic." Also they may be recombined as T/W and S/I, groups whose structures are isomorphic to the Abelian group Z2 X Z12 (and hence to one another). This paper shows the relationships among these four groups by means of, first, a miniature model consisting of a particular group isomorphic to D3, its anti-isomorphic group, and their two Abelian recombinations (analogous to the above), all acting on a system of two concentric equilateral triangles; then, second, an expanded model consisting of a particular group isomorphic to D12, its anti-isomorphic group, and their two Abelian recombinations, all acting on a system of two concentric circles with equispaced points located on their circumferences. This paper offers a geometric model for some of the central concepts in neo-Riemannian theory, in particular commuting groups (Lewin 1987), anti-isomorphisms (Lewin 1993), and contextual transposition and inversion as exemplified in the Schritt/Wechsel group (hereafter, S/W), conceived by Riemann (1880) and reformulated in explicitly group-theoretic