Unless I'm mistaken the neutral start is attached to the side of the
selector quadrant, it's also responsible for causing the ignition timing to
advance slightly when you select a gear, the same collection of copper
'wiping' contacts is also responsible for switching on the reversing lights.
This gear selector quadrant can actually move independantly of the switch,
and, unless its worn, it's likely that this positioning has moved a tiny
bit, which is all it takes to mess it up.
Hope this helps,
TTFN, Ken