Neural circuits are constructed from nonlinear building blocks, and not surprisingly overall circuit behavior is often strongly nonlinear. But neural circuits can also behave near linearly, and some circuits shift from linear to nonlinear behavior depending on stimulus conditions. Such control of nonlinear circuit behavior is fundamental to neural computation. Here, we study a surprising stimulus dependence of the responses of macaque On (but not Off) parasol retinal ganglion cells: these cells respond nonlinearly to spatial structure in some stimuli but near linearly to spatial structure in others, including natural inputs. We show that these differences in the linearity of the integration of spatial inputs can be explained by a shift in the balance of excitatory and inhibitory synaptic inputs that originates at least partially from adaptation in the cone photoreceptors. More generally, this highlights how subtle asymmetries in signaling - here in the cone signals - can qualitatively alter circuit computation.