The issue of the relation of eye position to perceptual reversals of the ambiguous figure of the 'Necker cube' dates back to Necker's original article [L.A. Necker (1832) The London & Edinburgh Philosophical Magazine and Journal of Science, 1, 329-337]. Despite the investigations of many distinguished psychophysicists since then, the question of whether perceptual switching is a cause or a consequence of associated changes in eye position has remained a matter of debate. In the present study we overcame methodological problems that have bedevilled many previous studies. We avoided any instruction that could interfere with the human subjects' free viewing of the Necker cube, tracked the eye position precisely and used biased versions of the cube that produced unambiguous percepts to determine how each subject actually looked at the cube. We show that, under these free-viewing conditions, there is a close link between the perception of the Necker cube and eye position. The average eye position of most subjects is at an extreme value at about the time when the subject's perception switches. From the biased cube trials we can infer that the polarity of the extreme corresponds to the percept which the subject had before the switch. These data indicate a bidirectional coupling between eye position and perceptual switching so that, after a subject's perceptual state changes, their eye position shifts to view the newly established percept. When the eye position approaches the corresponding extreme, the percept, in turn, becomes more and more likely to switch. This result suggests that the changed eye position itself might provide a negative feedback signal that suppresses the percept.