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We prove theorems showing a duality between the surface curvatures of three-dimensional objects and the existence of symmetry axes. More precisely, we prove that, given a surface, for each maximum or minimum of the principle curvature along a line of curvature, there is a symmetry axis terminating at this point. Moreover, such points are generically the only points at which these axes can terminate. These theorems generalize results obtained by Leyton for two-dimensional objects.