We examined two ways in which the neural control system for eye-head saccades constrains the motion of the eye in the head. The first constraint involves Listing's law, which holds ocular torsion at zero during head-fixed saccades. During eye-head saccades, does this law govern the eye's motion in space or in the head? Our subjects, instructed to saccade between space-fixed targets with the head held still in different positions, systematically violated Listing's law of the eye in space in a way that approximately, but not perfectly, preserved Listing's law of the eye in head. This finding implies that the brain does not compute desired eye position based on the desired gaze direction alone but also considers head position. The second constraint we studied was saturation, the process where desired-eye-position commands in the brain are "clipped" to keep them within an effective oculomotor range (EOMR), which is smaller than the mechanical range of eye motion. We studied the adaptability of the EOMR by asking subjects to make head-only saccades. As predicted by current eye-head models, subjects failed to hold their eyes still in their orbits. Unexpectedly, though, the range of eye-in-head motion in the horizontal-vertical plane was on average 31% smaller in area than during normal eye-head saccades, suggesting that the EOMR had been reduced by effort of will. Larger reductions were possible with altered visual input: when subjects donned pinhole glasses, the EOMR immediately shrank by 80%. But even with its reduced EOMR, the eye still moved into the "blind" region beyond the pinhole aperture during eye-head saccades. Then, as the head movement brought the saccade target toward the pinhole, the eyes reversed their motion, anticipating or roughly matching the target's motion even though it was still outside the pinhole and therefore invisible. This finding shows that the backward rotation of the eye is timed by internal computations, not by vision. When subjects wore slit glasses, their EOMRs shrank mostly in the direction perpendicular to the slit, showing that altered vision can change the shape as well as the size of the EOMR. A recent, three-dimensional model of eye-head coordination can explain all these findings if we add to it a mechanism for adjusting the EOMR.