Binocular rivalry and cross-orientation suppression are well-studied forms of competition in visual cortex, but models of these two types of competition are in tension with one another. Binocular rivalry occurs during the presentation of dichoptic grating stimuli, where two orthogonal gratings presented separately to the two eyes evoke strong alternations in perceptual dominance. Cross-orientation suppression occurs during the presentation of plaid stimuli, where the responses to a component grating presented to both eyes is weakened by the presence of a superimposed orthogonal grating. Conventional models of rivalry that rely on strong competition between orientation-selective neurons incorrectly predict rivalry between the components of plaids. Lowering the inhibitory weights in such models reduces rivalry for plaids, but also reduces it for dichoptic gratings. Using an exhaustive grid search, we show that this problem cannot be solved simply by adjusting the parameters of the model. Instead, we propose a robust class of models that rely on ocular opponency neurons, previously proposed as a mechanism for efficient stereo coding, to yield rivalry only for dichoptic gratings, not for plaids. This class of models reconciles models of binocular rivalry with the divisive normalization framework that has been used to explain cross-orientation. Our model makes novel predictions that we confirmed with psychophysical tests.