We report the observation of nonlinear interactions in quadratic nonlinear crystals having a geometrically twisted susceptibility pattern. The quasi-angular-momentum of these crystals is imprinted on the interacting photons during the nonlinear process so that the total angular momentum is conserved. These crystals affect three basic physical quantities of the output photons: energy, translational momentum, and angular momentum. Here we study the case of second-order harmonic vortex beams, generated from a Gaussian pump beam. These crystals can be used to produce multidimensional entanglement of photons by angular momentum states or for shaping the vortex's structure and polarization.