Abstract To understand the complexity of stimulus selectivity in primary visual cortex (V1), models constructed to match observed responses to complex time-varying stimuli, instead of to explain responses to simple parametric stimuli, are increasingly used. While such models often can more accurately reflect the computations performed by V1 neurons in more natural visual environments, they do not by themselves provide insight into established measures of V1 neural selectivity such as receptive field size, spatial frequency tuning and phase invariance. Here, we suggest a series of analyses that can be directly applied to encoding models to link complex encoding models to more interpretable aspects of stimulus selectivity, applied to nonlinear models of V1 neurons recorded in awake macaque in response to random bar stimuli. In linking model properties to more classical measurements, we demonstrate several novel aspects of V1 selectivity not available to simpler experimental measurements. For example, we find that individual spatiotemporal elements of the V1 models often have a smaller spatial scale than the overall neuron sensitivity, and that this results in non-trivial tuning to spatial frequencies. Additionally, our proposed measures of nonlinear integration suggest that more classical classifications of V1 neurons into simple versus complex cells are spatial-frequency dependent. In total, rather than obfuscate classical characterizations of V1 neurons, model-based characterizations offer a means to more fully understand their selectivity, and provide a means to link their classical tuning properties to their roles in more complex, natural, visual processing. Significance statement Visual neurons are increasingly being studied with more complex, natural visual stimuli, with increasingly complex models necessary to characterize their response properties. Here, we describe a battery of analyses that relate these more complex models to classical characterizations. Using such model-based characterizations of V1 neurons furthermore yields several new insights into V1 processing not possible to capture in more classical means to measure their visual selectivity.