What are the functions of the parietal and prefrontal cortices and how do these regions interact to achieve their objectives? How does context change plans for action? Are 'visual' and 'motor' well-defined attributes of neuronal activity in the association cortex? Recent studies of the saccadic system in monkeys have revealed two sets of paradoxical data: in parietal cortex, activity has visual timing but motor direction; in prefrontal cortex, activity has motor timing but visual direction. Analyzing the prefrontal and parietal data together leads to surprising insights. It appears that these paradoxical activities are intermediates in a parietal-prefrontal-parietal loop that has a rapid turnaround, and that a possibly prefrontal context-contingent signal switches sensorimotor transformations in parietal cortex.