The majority of synapses in primary visual cortex mediate excitation between nearby neurons, yet the role of local recurrent connections in visual processing remains unclear. We propose that these connections are responsible for the spatial-phase invariance of complex-cell responses. In a network model with selective cortical amplification, neurons exhibit simple-cell responses when recurrent connections are weak and complex-cell responses when they are strong, suggesting that simple and complex cells are the low- and high-gain limits of the same basic cortical circuit. Given the ubiquity of invariant responses in cognitive processing, the recurrent mechanism we propose for complex cells may be widely applicable.