At low temperatures the phase diagram for the quantum Hall effect has a powerful symmetry arising from the law of corresponding states. This symmetry gives rise to an infinite order discrete group, which is a generalization of Kramers-Wannier duality for the two-dimensional Ising model. The duality group, which is a subgroup of the modular group, is analyzed and it is argued that there is a quantitative difference between a situation in which the spin splitting of electron energy levels is comparable to the cyclotron energy and one in which the spin splitting is much less than the cyclotron energy. In the former case the group of symmetries is larger than in the latter case. These duality symmetries are used to constrain the scaling functions of the theory and, under an assumption of complex meromorphicity, a unique functional form is obtained for the crossover of the conductivities between Hall states as a function of the external magnetic field. This analytic form is shown to give good agreement with experimental data. The analysis requires a consideration of the way in which longitudinal resistivities are extracted from the experimentally measured longitudinal resistances and a method is proposed for determining the correct normalization for the former.