Stretching (or compressing) a face by a factor of two has no effect on its recognition as assessed by the speed and accuracy of judging whether the face is that of a celebrity (Hole, 2002). This invariance has stood as a challenge to all contemporary accounts of the relation between neurocomputational measures of face similarity and face recognition. We extend the documentation of strong invariance over compression to a factor of four and show that the deformation so produced is sufficiently great that the resultant image is as similar to markedly different faces—even those differing in race, sex, and expression-- as it is to the original face. The invariance to face compression is readily witnessed with less familiar celebrities and unfamiliar faces ruling out a role of exposure to transformed images of particular faces through depth rotation or viewing pictures at varied viewing angles. We additionally discount the possibility that faces are “un-stretched” by warping them to an average face. Instead, we suggest that the percept of an elongated face provides a signal for the shrink-wrapping of receptive fields to conform to an attended object, a phenomenon witnessed in single unit activity in the macaque by Moran and Desimone (1985) which may serve, more generally, as the underlying neural mechanism for object-based attention.