Within the huge family of Heusler compounds only a few quaternary derivatives are known that crystallize in the F43m space group. In this work, the yet unreported compounds CoRhMnZ (Z = Ga, Sn, Sb) and the alloy Co(0.5)Rh(1.5)MnSb were investigated in detail by experimental techniques and theoretical methods. The ab initio calculations predict the CoRhMnZ compounds to be half-metallic ferromagnets or to be close to the half-metallic ferromagnetic state. Calculations of the elastic constants show that the cubic structure is stable in compounds containing Mn. Both calculations and experiment reveal that Mn cannot be exchanged by Fe (CoRhFeGa). The low temperature magnetization of the compounds is in the range of 3.4-5.5 μ(B) depending on the composition. The best agreement between experiment and calculation has been achieved for CoRhMnSn (5 μ(B)). The other compounds are also cubic but tend to anti-site disorder. Compared to Co(2)MnSn it is interesting to note that the magnetic properties and half-metallicity are preserved when replacing one of the 'magnetic' Co atoms by a 'non-magnetic' Rh atom. This allows us to increase the spin-orbit interaction at one of the lattice sites while keeping the properties as a precondition for applications and physical effects relying on a large spin-orbit interaction. The Curie temperatures were determined from measurements in induction fields of up to 1 T by applying molecular field fits respecting the applied field. The highest Curie temperature was found for CoRhMnSn (620 K) that makes it, together with the other well defined properties, attractive for above room temperature spintronic applications.