In addressing the scientific study of consciousness, Crick and Koch state, “It is probable that at any moment some active neuronal processes in your head correlate with consciousness, while others do not: what is the difference between them?” (1998, p. 97). Evidence from electrophysiological and brain-imaging studies of binocular rivalry supports the premise of this statement and answers to some extent, the question posed. I discuss these recent developments and outline the rationale and experimental evidence for the interhemispheric switch hypothesis of perceptual rivalry. According to this model, the perceptual alternations of rivalry reflect hemispheric alternations, suggesting that visual consciousness of rivalling stimuli may be unihemispheric at any one time (Miller et al., 2000). However, in this paper, I suggest that interhemispheric switching could involve alternating unihemispheric attentional selection of neuronal processes for access to visual consciousness. On this view, visual consciousness during rivalry could be bihemispheric because the processes constitutive of attentional selection may be distinct from those constitutive of visual consciousness. This is a special case of the important distinction between the neuronal correlates and constitution of visual consciousness.