Imitation Learning (IL) is an effective framework to learn visuomotor skills from offline demonstration data. However, IL methods often fail to generalize to new scene configurations not covered by training data. On the other hand, humans can manipulate objects in varying conditions. Key to such capability is hand-eye coordination, a cognitive ability that enables humans to adaptively direct their movements at task-relevant objects and be invariant to the objects absolute spatial location. In this work, we present a learnable action space, Hand-eye Action Networks (HAN), that can approximate humans hand-eye coordination behaviors by learning from human teleoperated demonstrations. Through a set of challenging multi-stage manipulation tasks, we show that a visuomotor policy equipped with HAN is able to inherit the key spatial invariance property of hand-eye coordination and achieve zero-shot generalization to new scene configurations. Additional materials available at https://sites.google.com/stanford.edu/han