Pyramidal cells and interneurons expressing parvalbumin (PV), somatostatin (SST), and vasoactive intestinal peptide (VIP) show cell-type-specific connectivity patterns leading to a canonical microcircuit across cortex. Experiments recording from this circuit often report counterintuitive and seemingly contradictory findings. For example, the response of SST cells in mouse V1 to top-down behavioral modulation can change its sign when the visual input changes, a phenomenon that we call response reversal. We developed a theoretical framework to explain these seemingly contradictory effects as emerging phenomena in circuits with two key features, interactions between multiple neural populations and a nonlinear neuronal input-output relationship. Furthermore, we built a cortical circuit model which reproduces counterintuitive dynamics observed in mouse V1. Our analytical calculations pinpoint connection properties critical to response reversal, and predict additional novel types of complex dynamics that could be tested in future experiments.