The interconnected areas of the visual system work together to find object boundaries in visual scenes. Primary visual cortex (V1) mainly extracts oriented luminance boundaries, while secondary visual cortex (V2) also detects boundaries defined by differences in texture. How the outputs of V1 neurons are combined to allow for the extraction of these more complex boundaries in V2 is as of yet unclear. To address this question, we probed the processing of orientation signals in single neurons in V1 and V2, focusing on response dynamics of neurons to patches of oriented gratings and to combinations of gratings in neighboring patches and sequential time frames. We found two kinds of response dynamics in V2, both of which were different from those of V1 neurons. While V1 neurons in general preferred one orientation, one subpopulation of V2 neurons (“transient”) showed a temporally dynamic preference, resulting in a preference for changes in orientation. The second subpopulation of V2 neurons (“sustained”) responded similarly to V1 neurons, but with a delay. The dynamics of nonlinear responses to combinations of gratings reinforced these distinctions: the dynamics enhanced the preference of V1 neurons for continuous orientations and the preference of V2 transient neurons for discontinuous ones. We propose that transient neurons in V2 perform a differentiation operation on the V1 input, both spatially and temporally, while the sustained neurons perform an integration operation. We show that a simple feedforward network with delayed inhibition can account for the temporal but not for the spatial differentiation operation.