It has been proposed that the capacity to code the 'like me' analogy between self and others constitutes a basic prerequisite and a starting point for social cognition. It is by means of this self/other equivalence that meaningful social bonds can be established, that we can recognize others as similar to us, and that imitation can take place. In this article I discuss recent neurophysiological and brain imaging data on monkeys and humans, showing that the 'like me' analogy may rest upon a series of 'mirror-matching' mechanisms. A new conceptual tool able to capture the richness of the experiences we share with others is introduced: the shared manifold of intersubjectivity. I propose that all kinds of interpersonal relations (imitation, empathy and the attribution of intentions) depend, at a basic level, on the constitution of a shared manifold space. This shared space is functionally characterized by automatic, unconscious embodied simulation routines.