During binocular rivalry, perception alternates between two different images presented one to each eye. At any moment, one image is visible, dominant, while the other is invisible, suppressed. Alternations in perception during rivalry could involve competition between eyes, eye-rivalry, or between images, image-rivalry, or both. We measured response criteria, sensitivities, and thresholds to brief contrast increments to one of the rival stimuli in conventional rivalry displays and in a display in which the rival stimuli swapped between the eyes every 333 ms-swap rivalry-that necessarily involves image rivalry. We compared the sensitivity and threshold measures in dominance and suppression to assess the strength of suppression. We found that response criteria are essentially the same during dominance and suppression for the two sorts of rivalry. Critically, we found that swap-rivalry suppression is weak after a swap and strengthens throughout the swap interval. We propose that image rivalry is responsible for weak initial suppression immediately after a swap and that eye rivalry is responsible for the stronger suppression that comes later.