The magnetization reversal processes in magnetic bilayers with individual uniaxial anisotropies have been studied, both theoretically and experimentally, to analyze the possible existence of inverted hysteresis loops, that is, with negative remanent magnetization ${(M}_{r}).$ Kerr effect measurements in amorphous ${\mathrm{YCo}}_{2}/{\mathrm{YCo}}_{2}$ bilayers and alternating gradient magnetometry in polycrystalline FeNi/FeNi samples reveal that ${M}_{r}<0$ can be observed for certain directions of the applied magnetic field in the sample plane. This property has also been found in CoNbZr films annealed under an applied field. Our theoretical approach shows that the behavior of these magnetic heterogeneous systems with two coupled uniaxial anisotropies can be understood in terms of two competing effective anisotropies, one biaxial (with ${K}_{\mathrm{biax}})$ and one uniaxial (with ${K}_{\mathrm{uniax}}).$ In particular, a phase diagram has been deduced for the conditions on ${K}_{\mathrm{biax}}$ and ${K}_{\mathrm{uniax}}$ that can produce negative remanence. This description indicates that, under those anisotropy conditions, inverted hysteresis loops can be observed for an applied field close to the hard axis of the effective uniaxial anisotropy, when magnetization reversal is driven by rotations and not by domain nucleation and wall movement. To consider the real situation in a ${\mathrm{YCo}}_{2}/{\mathrm{YCo}}_{2}$ bilayer sample, the predictions of this phenomenological model have been further improved by micromagnetic calculations, which are in very good agreement with the magneto-optical measurements.