Euler's Rotation Theorem states that any orientation-preserving isometry (rigid motion) of a sphere is equivalent to a rotation by some amount about some axis. As the earth wobbles randomly in the animation below, the red line indicates the axis around which the earth must be rotated from its current position to regain its starting position. Thus the isometry given by movement of the earth from its starting position to its current position is the opposite rotation (again about the red line). To see the rotation back to the starting position, click anywhere on the figure. Click again to resume the random wobbling.

One way to see this theorem as plausible is to think about the orbit of a point under the action of the motion (rotation) from the starting orientation to the current orientation. You can see some points from the orbit of Chicago by incrementing the value of the box below. Note that if the value is set to 1, only the zeroeth iteration (the starting position of Chicago) will be shown.

Don't see anything? Most likely your browser isn't set up for WebGL. Maybe get.webgl.org will be helpful. If you see the spinning cube on that website but don't see anything here, do complain.

Source available on GitHub. Created with three.js. World map from User:Koba-chan on Wikimedia Commons, CC-BY-SA-3.0.