OK so resetting the smc hasn't worked (damn!)
The screen had been failing to come on, then flicking on, then off again. Now it's permanently off. The Mac works fine and attaching an external display works a treat.
So I guess it's either the converter board or the backlight, although there is absolutely NO image, not even a very faint one which I would expect to see if it was the backlight so I'm guessing the inverter board.
Any other guesses?