Canonical symmetries determine classes of hamiltonians in analogy with the way gauge symmetries specify sectors of Hilbert space. By investigating Z(2) gauge and spin models we show that these two concepts are dual to one another for D = 2 space dimensions. We argue that identifying the mass-gap with the appropriate symmetry excitation of either kind is of practical importance in finite-size scaling calculations. Applying these ideas to the Z(2) gauge and matter theory we suggest a symmetry characterization of its phase boundaries, identifying them with lines along which the self-energy of an external charge (or canonical excitation) vanishes. We obtain interesting numerical results which exhibit Ising characteristics in 2 + 1 dimensions.