Flexible visually guided behavior requires visual generalization, mapping diverse inputs to a common inference, and behavioral generalization, mapping one inference to multiple actions. Here, we tested whether visual neurons support both by maintaining a relatively fixed mapping between visual information amid many variations in stimuli and actions. We recorded neuronal populations in visual cortex in two male monkeys while they estimated the curvature of random 3D objects and flexibly mapped those judgments to different eye movements. Although visual responses varied substantially across objects, perceptual judgments were best explained by a common, shape-general readout of activity in V1 and V4, rather than by shape-specific strategies. V4 population activity additionally encoded impending saccades while maintaining a stable representation of stimulus curvature, allowing a single perceptual estimate to be linked to multiple behavioral outputs. Together, our findings suggest that visual cortex populations contain the ingredients necessary to implement multiple forms of flexible generalization.