Uncovering which feature combinations are encoded by visual units is critical to understanding how images are transformed into representations that support recognition. While existing feature visualization approaches typically infer a unit's most exciting images, this is insufficient to reveal the manifold of transformations under which responses remain invariant, which is critical to generalization in vision. Here we introduce Stretch-and-Squeeze (SnS), a model-agnostic, gradient-free framework to systematically characterize a unit's maximally invariant stimuli, and its vulnerability to adversarial perturbations, in both biological and artificial visual systems. SnS frames these transformations as bi-objective optimization problems. To probe invariance, SnS seeks image perturbations that maximally alter (stretch) the representation of a reference stimulus in a given processing stage while preserving unit activation downstream (squeeze). To probe adversarial sensitivity, stretching and squeezing are reversed to maximally perturb unit activation while minimizing changes to the upstream representation. Applied to CNNs, SnS revealed invariant transformations that were farther from a reference image in pixel-space than those produced by affine transformations, while more strongly preserving the target unit's response. The discovered invariant images differed depending on the stage of the image representation used for optimization: pixel-level changes primarily affected luminance and contrast, while stretching mid- and late-layer representations mainly altered texture and pose. By measuring how well the hierarchical invariant images obtained for L2 robust networks were classified by humans and other observer networks, we discovered a substantial drop in their interpretability when the representation was stretched in deep layers, while the opposite trend was found for standard models.