The propagation of electromagnetic plane waves in a material with simultaneous mirror-conjugated and racemic chirality (SMCRC) characteristics is investigated. General conditions for negative phase velocity (NPV) (i.e., phase velocity directed opposite to the time-averaged Poynting vector) are derived for both unirefringent and birefringent propagation. Through numerical studies, it is demonstrated that NPV propagation arises provided that the magnitude of the magnetoelectric constitutive parameter is sufficiently large compared with the magnitudes of dielectric and magnetic constitutive parameters. However, the relative magnitude of the magnetoelectric constitutive parameter has little bearing upon the directions, which support NPV propagation. The propensity for NPV propagation is much enhanced through incorporating dielectric and magnetic constitutive parameters, which are negative-real. A wide range of constitutive parameter values are considered in order to accommodate the possibilities offered by the fabrication of artificial SMCRC materials.