Onsager crosses are hard nonconvex bodies formed by rigidly connecting three elongated rods, equally thick but not necessarily equally long, to form perpendicular crosses. We study the phase behavior of systems of such particles, focusing on their ability to form spatially homogeneous orientationally ordered phases with a symmetry lower than that of the standard uniaxial nematic. We treat these systems in the Onsager, second virial coefficient, approximation. We apply bifurcation analysis to build up a global picture of the phase diagram, which is then refined using approximate numerical calculations. Finally, we generalize the Gaussian approximation for the nematic orientational distribution function, to deal with the more ordered phases encountered here, and compare with the results from the previous techniques to see whether it is feasible to reliably predict the phase diagrams from a computationally cheaper technique.