To understand the brain areas associated with visual awareness and their anatomical interconnections, we studied binocular rivalry with functional magnetic resonance imaging (fMRI) and diffusion tensor imaging (DTI). Binocular rivalry occurs when one image is viewed by one eye and a different image by the other; it is experienced as perceptual alternations between the two images. Our first experiment addressed problems with a popular comparison condition, namely permanent suppression, by comparing rivalry with binocular fusion instead. We found an increased fMRI signal in right frontal, parietal, and occipital regions during rivalry viewing. The pattern of neural activity differed from findings of permanent suppression comparisons, except for adjacent activity in the right superior parietal lobule. This location was near fMRI signal changes related to reported rivalry alternations in our second experiment, indicating that neighbouring areas in the right parietal cortex may be involved in different components of rivalry. In our second experiment, we used probabilistic tractography to detect white matter fibres between right-hemispheric areas that showed event-related fMRI signal changes time-locked to reported perceptual alternations during rivalry viewing. Most of these functionally defined areas were linked by probabilistic fibre tracts, some of which followed long-distance connections such as the inferior occipitofrontal fasciculus. Corresponding anatomical pathways might mediate communication within the functional network associated with changes in conscious perception during binocular rivalry.