I was helping out someone in IRC earlier with a quaternion problem. He was facing a mental hurdle that I remember facing myself, and I'm thinking that perhaps other people have hit it as well. It's the difference between "rotation" and "orientation."

Contrary to popular usage, the two words do not mean the same thing. The difference is that orientation is rotation relative to some 'origin' orientation.

Orientation is a property, while rotation is an operation. It's like the difference between position and distance - "3 miles" is a distance, but you can't say "this point is 3 miles" - it doesn't make sense unless you say 3 miles from what.

So you can get stuck if you try and convert something like a quaternion (which represents a rotation) into a direction (which is a property). You can't do it unless you define your spatial system first - to say that (0, 0, 1) is 'unrotated.' Things just don't make sense otherwise.