It wouldn't take much of a tweak on one of the doors to cause a little bind at the pivot point - probably less than the eye can see.
My first thought as I was reading your post was that the head was a little out-of-square, causing one side to bind. But, as you say, the problem follows the door so the problem must be with the door.
How does it shave? Do both sides shave the same, or is one more aggressive than the other?
Brad - OGA
You must be willing to do the things today others won't do . . .
In order to have the things tomorrow others won't have. - Les Brown