Human eye-gaze is a powerful stimulus, drawing the observer's attention to places and objects of interest to someone else ('eye-gaze following'). The largely homogeneous eyes of monkeys, compromising the assessment of eye-gaze by conspecifics from larger distances, explain the absence of comparable eye-gaze following in these animals. Yet, monkeys are able to use peer head orientation to shift attention ('head-gaze following'). How similar are monkeys' head-gaze and human eye-gaze following? To address this question, we trained rhesus monkeys to make saccades to targets, either identified by the head-gaze of demonstrator monkeys or, alternatively, identified by learned associations between the demonstrators' facial identities and the targets (gaze versus identity following). In a variant of this task that occurred at random, the instruction to follow head-gaze or identity was replaced in the course of a trial by the new rule to detect a change of luminance of one of the saccade targets. Although this change-of-rule rendered the demonstrator portraits irrelevant, they nevertheless influenced performance, reflecting a precise redistribution of spatial attention. The specific features depended on whether the initial rule was head-gaze or identity following: head-gaze caused an insuppressible shift of attention to the target gazed at by the demonstrator, whereas identity matching prompted much later shifts of attention, however, only if the initial rule had been identity following. Furthermore, shifts of attention prompted by head-gaze were spatially precise. Automaticity and swiftness, spatial precision and limited executive control characterizing monkeys' head-gaze following are key features of human eye-gaze following. This similarity supports the notion that both may rely on the same conserved neural circuitry.