The influence of cognitive context on orienting behaviour can be explored using the mixed memory-prosaccade, memory-antisaccade task. A symbolic cue, such as the colour of a visual stimulus, instructs the subject to make a brief, rapid eye movement (a saccade) either towards the stimulus (prosaccade) or in the opposite direction (antisaccade). Thus, the appropriate sensorimotor transformation must be switched on to execute the instructed task. Despite advances in our understanding of the neuronal processing of antisaccades, it remains unclear how the brain selects and computes the sensorimotor transformation leading to an antisaccade. Here we show that area LIP of the posterior parietal cortex is involved in these processes. LIP's population activity turns from the visual direction to the motor direction during memory-antisaccade trials. About one-third of the visual neurons in LIP produce a brisk, transient discharge in certain memory-antisaccade trials. We call this discharge 'paradoxical' because its timing is visual-like but its direction is motor. The paradoxical discharge shows, first, that switching occurs already at the level of visual cells, as previously proposed by Schlag-Rey and colleagues; and second, that this switching is accomplished very rapidly, within 50 ms from the arrival of the visual signals in LIP.