How do I measure how far left/right and up/down an object is looking in terms of rotation?

I need to know how far left or right and up or down the face of an object is looking. This would need to be out of a 100% of some value, such as 360, from the point of origin.

What I'm trying to do is trying to map a Vive controllers rotation to movement of the mouse, so that rotating the controller from it's origin point around the x axis moves the mouse from one side of the screen to the other as a full rotation is completed.

Now I can already move the mouse incrementally based a change in a value out of a 100%, but I need to be able to figure out the controllers movements along the two axis representing up/down and left/right. These would be X and Y presumably.