In this talk, I will discuss the degree to which Kraichnan-Leith-Batchelor (KLB) self-similar inertial range phenomenology describes inverse energy cascades in a family of two-dimensional turbulence models. Using results from numerical simulations, I will show that the energy-cascading inertial range is only self-similar and described by KLB phenomenology for a limited range of two-dimensional fluid models and specific choices of forcing and dissipation. In some models, deviations from self-similarity occur due to coherent vortex formation, which causes energy spectra to steepen past the KLB prediction. In these cases, decomposing the vorticity field into coherent and turbulent background components reveals that the turbulent background retains the KLB similarity spectrum. For other models, turbulence closure calculations indicate that the self-similar energy cascade is associated with energy flux toward small scales, which is not physically realizable. We thus do not expect to observe a self-similar energy cascade in these models under any conditions, and this is confirmed by numerical simulations. Despite wide variations in their characteristics, such as the prevalence of coherent structures, the energy cascades under study all exhibit a scaling behaviour known as extended self-similarity, which appears to be more fundamental than KLB scaling.