Humans and other animals can identify objects by active touch, requiring the coordination of exploratory motion and tactile sensation. Both the motor strategies and neural representations employed could depend on the subject's goals. We developed a shape discrimination task that challenged head-fixed mice to discriminate concave from convex shapes. Behavioral decoding revealed that mice did this by comparing contacts across whiskers. In contrast, a separate group of mice performing a shape detection task simply summed up contacts over whiskers. We recorded populations of neurons in the barrel cortex, which processes whisker input, and found that individual neurons across the cortical layers encoded touch, whisker motion, and task-related signals. Sensory representations were task-specific: during shape discrimination, but not detection, neurons responded most to behaviorally relevant whiskers, overriding somatotopy. Thus, sensory cortex employs task-specific representations compatible with behaviorally relevant computations.