Without reading up on this, I imagine the earth's orbit would trace a more or less level path around the sun. If I was to imagine this as a spiral I would have to first assume the solar system as a whole is moving through space in a direction perpendicular to the earth's orbit. This would then I suppose trace out a spiral.
If the movement of the solar system as a whole was not perpendicular to the orbit of the earth then the spiral would not be as I imagine a spiral to be - which is like a slinky.
I strongly suspect my conception of this is waaaay off. The answer might lie in topology, but it's hard for me to know because I'm not on top of (
) all the dimensions in the universe which would need to be taken into account for such a model to be formulated.