We argue that tetragonal CuO (T-CuO) has the potential to finally settle long-standing modeling issues for cuprate physics. We compare the one-hole quasiparticle (qp) dispersion of T-CuO to that of cuprates, in the framework of the strongly correlated (U-dd -> infinity) limit of the three-band Emery model. Unlike in CuO2, magnetic frustration in T-CuO breaks the C-4 rotational symmetry and leads to strong deviations from the Zhang-Rice singlet picture in parts of the reciprocal space. Our results are consistent with angle-resolved photoemission spectroscopy data but in sharp contradiction to those of a one-band model previously suggested for them. These differences identify T-CuO as an ideal material to test a variety of scenarios proposed for explaining cuprate phenomenology.