Binocular coordination of eye movements is essential for stereopsis (depth perception) and to prevent double vision. More than a century ago, Hering and Helmholtz debated the neural basis of binocular coordination. Helmholtz believed that each eye is controlled independently and that binocular coordination is learned. Hering believed that both eyes are innervated by common command signals that yoke the eye movements (Hering's law of equal innervation). Here we provide evidence that Hering's law is unlikely to be correct. We show that premotor neurons in the paramedian pontine reticular formation that were thought to encode conjugate velocity commands for saccades (rapid eye movements) actually encode monocular commands for either right or left eye saccades. However, 66% of the abducens motor neurons, which innervate the ipsilateral lateral rectus muscle, fire as a result of movements of either eye. The distribution of sensitivity to ipsilateral and contralateral eye movements across the abducens motor neuron pool may provide a basis for learning binocular coordination in infancy and adapting it throughout life.