My nearside door mirror, instead of dipping when reverse is selected, has started 'hunting'.

It moves backwards and forwards from left to right continuously until reverse is deselected

at which point, it returns to its normal position. Is there a cure?

Just to clarify, the mirror hunts from left to right, not up and down when reverse is selected?

Assuming you have the ideal normal position set in the memory, try adjusting the mirror left/right (past the limits it goes to when reverse is selected) and then hit the Memory button to bring it back to where it should be. Do this several times and then try reverse again. My theory is the tracks on the pots that give feedback of the mirror position have become dirty so the ECU that controls the mirror is trying to find that spot where it gets the correct feedback.

Hopefully this will fix it, there may be a recalibration procedure for the mirrors that i'm not aware of (usually after a battery disconnect/flat battery) that might also fix things, other than that i think you'll need a code reader to see if any codes are present and then maybe investigate whichever ECU (General Electronic Module perhaps) controls the mirrors.