We demonstrate, both experimentally and theoretically, a method that allows us to modify the spatial properties of photon pairs produced by the process of spontaneous parametric downconversion (SPDC) with an apertured, zeroth-order Bessel-Gauss beam. Concentrating on the use of a half-plane aperture, we show that the phasematching conditions for non-paraxial pump beams can be controlled by simply changing the aperture orientation. Partially blocking the pump beam, leads to a selection of either one of two individual cones that form a dual-cone SPDC angular spectrum, or a portion of both of them. We can likewise determine the shape and orientation of the conditional angular spectrum. This control of the SPDC spatial properties could be useful for quantum information processing protocols.