A lack of spatial inversion symmetry gives rise to a variety of unconventional physics, from noncollinear order and Skyrmion lattice phases in magnetic materials to topologically protected surface states in certain band insulators to mixed-parity pairing states in superconductors. The search for exotic physics in such materials is largely limited by a lack of candidate materials and often by difficulty in obtaining crystals. Here, we report the single-crystal growth and physical properties of the noncentrosymmetric tungsten aluminide cage compounds Al4W and Al5W, alongside related molybdenum aluminides in which spin-orbit coupling should be significantly weaker. All compounds are nonmagnetic metals. Their high conductivities suggest the opportunity to find superconductivity at lower temperatures, while the limits we can place on their transition temperatures suggest that any superconductivity may be expected to exhibit significant parity mixing.