In this article we investigate the connection between the chirality of interacting vortices and the appearance of radial structures in nonlinear wave mixing. Depending on the signs of their topological charges, the nonlinear mixing of optical vortices may produce a radial-angular coupling that generates a finite superposition of pure Laguerre-Gaussian modes carrying the resultant topological charge and a finite spectrum of radial orders. These radial modes evolve with different Gouy phases that determine the transformation from a hollow intensity distribution in the near field to a finite ring structure in the far field pattern. In this sense, we interpret the appearance of radial modes in nonlinear wave mixing as a diffraction of the up-converted beam through the effective amplitude-phase mask created by the pump beams. This interpretation is supported by comparison between the images produced by the nonlinear process and purely diffractive measurements with a spatial light modulator that mimics the amplitude-phase modulation produced in nonlinear wave mixing.