Sounds like its your ignition switch.  VW provides power to the headlights via two paths.  If you pull back on the lever, power is supplied directly to the brights, without going through the ignition switch or headlight switch.  The headlight switch is connected to the "X" terminal of the ignition switch, which is supplied with power when the key is on, but not in the start position.  It sounds like your X contact in the ignition switch is flakey.  Do your windshield wipers work?  I think they are wired (via a relay) to the X terminal too.

I just went through this on a '90 Cabriolet.  Some @#$% idiot ran a wire from the X terminal (which is UNFUSED) to the trunk (where an amplifier had been installed), then left the bare end of the wire just hanging there!  I got the car cheap, after a shop quoted the PO $2000 to fix the damage caused by the short!

Bill

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This thing is baffling me. Headlights do not come on when the engine is running. I can hold the bright lever back and the brights work. Funny idiosyncracy: if I turn the headlight switch on, turn the ignition key to accessory position, then the headlights come on. When I keep turning the key toward start, they go off, then just before getting to start position they come back on, then when the engine has started and key is resting, they go off....but the brights work if I hold the lever back .....so.... does not appear to be the "switch". Any ideas?

I have all the parts to put the relays in and will do that this weekend but am confused about how that may influence this "current" problem.

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