I was there once and we went south. True south. And after about 20 miles looked down, checked what line of longitude we were on (somewhere heading towards Russia) took a right 90 for a few minutes, then when we hit the longitude that runs throught Andoya, Norway, took a left 90 and were on our way. We had intertials and GPS.
The correct way to do it is using Grid Nav. Our navigator hadn't brushed up on it and truthfully, us pilots barely know what to call it much less do it. I imagine in a plane without a navigator the pilots would have to get more up to speed.
If you look at a chart of the polar regions there are blue (I think) squares on there that are superimposed over the normal latitude and longitude lines. By going to a special function of your inertials, you can basically fool your intertials into giving you steering based on those blue lines (the grid) and you fly around using N,E,S, and W directions that related to the blue grid squares rather than the global N,E,S,W directions.
I don't know if the guys going North Amercia to Europe do grid nav or not, seems like if you are going straight shot it wouldn't be worth the trouble, especially with GPS. Military uses it for doing things like tracking a submarine up there. If you have to fly all around and accurately be able to come back to a position you need some kind of work around and Grid Nav is it.
The answer is a true heading of 180 with the autopilot coupled to the INS. However just for kicks, if you wanted to fly from true north pole to the magnetic north pole you'd fly a magnetic heading of 360, because the magnetic variation along this line is 180 degrees.
I was having consciousness trouble during a long range nav class and vaguely remember hearing the instructor talk about the ins being prone to errors in the North Pole. I guess I'll just avoid trips that go that way.