During binocular rivalry, one stimulus is visible (dominant), while the other stimulus is invisible (suppressed); after a few seconds, perception reverses. To determine whether these alternations involve competition between the eyes or between the images, we measured suppression depth to monocular probes. We did so in conventional rival stimuli and in rival stimuli swapping between the eyes at 1.5 Hz (both sorts of rivalry were shown either with or without 18-Hz flicker). The conventional conditions cause rivalry that could involve either competition between the eyes or between the images or both. The eye-swapping conditions cause rivalry that could involve competition between the images. Probes were either a small spot or a contrast increment to one of the rival stimuli. Using both yes–no and forced-choice procedures, we found that conventional conditions yielded large suppression depth and that eye-swapping conditions yielded small suppression depth. Weak suppression during image rivalry is consistent with conventional rivalry's involving competition between eyes and between images and flicker-and-swap rivalry's involving little, if any competition between eyes and mainly competition between images.