We present experimental evidence of transverse magnetic domains, previously observed only in nanostrips, in CoNi cylindrical nanowires with designed crystal symmetry and tailored magnetic anisotropy. The transverse domains are found together with more conventional vortex domains along the same cylindrical nanowire, denoting a bistable system with similar energies. The surface and the inner magnetization distribution in both types of domains are analyzed by photoemission electron microscopy with x-ray magnetic circular dichroism contrast, and hysteresis loop in individual nanowires are measured by magneto-optical Kerr effect. These experimental data are understood and compared with complementary micromagnetic simulations.