The handedness of Leonardo da Vinci is controversial. Although there is little doubt that many of his well-attributed drawings were drawn with the left hand, the hatch marks of the shading going downwards from left to right, it is not clear that he was a natural left-hander, there being some suggestion that he may have become left-handed as the result of an injury to his right hand in early adulthood. Leonardo's lateralisation may be illuminated by an obscure passage in his notebooks in which he describes crouching down to look into a dark cave, putting his left hand on his knee, and shading his eyes with his right hand. We carried out a questionnaire survey, using 33 written and photographic items, to find whether this behaviour was typical of right handers or left handers. In fact the "Leonardo task" showed almost no direct association with handedness, meaning that it contributes little to the immediate problem of elucidating Leonardo's handedness. However, the lateralisation of the task did relate to other aspects of behavioural laterality in surprisingly complex ways. This suggests that individual differences in handedness, and behavioural laterality in general, have a structural complexity which is not fully encompassed by simple measures of direction or degree of handedness.