Theoretical frameworks of human vision argue that object responses remain stable, or 'invariant', despite changes in viewing conditions that can alter object appearance but not identity. Here, in a major departure from previous approaches that have relied on two-dimensional (2-D) images to study object processing, we demonstrate that changes in an object's appearance, but not its identity, can lead to striking shifts in behavioral responses to objects. We used inverse multidimensional scaling (MDS) to measure the extent to which arrangements of objects in a sorting task were similar or different when the stimuli were displayed as scaled 2-D images, three-dimensional (3-D) augmented reality (AR) projections, or real-world solids. We were especially interested in whether sorting behavior in each display format was based on conceptual (e.g., typical location) versus physical object characteristics. We found that 2-D images of objects were arranged according to conceptual (typical location), but not physical, properties. AR projections, conversely, were arranged primarily according to physical properties such as real-world size, elongation and weight, but not conceptual properties. Real-world solid objects, unlike both 2-D and 3-D images, were arranged using multidimensional criteria that incorporated both conceptual and physical object characteristics. Our results suggest that object responses can be strikingly malleable, rather than invariant, with changes in the visual characteristics of the stimulus. The findings raise important questions about limits of invariance in object processing, and underscore the importance of studying responses to richer stimuli that more closely resemble those we encounter in real-world environments.