What is the direction of a rotation?

We usually talk about rotations in the clockwise or counterclockwise direction. But if a rotation is just a function defined on the space, then it is all about points and their images, and there is no "direction" involved.

For example, consider the unique rotation about $$(0,0)$$ that sends the point $$(1,0)$$ to the point $$(0,1)$$ in the plane. Is it in the clockwise or the counterclockwise direction ?

We could say it's counterclockwise because it's a 90° turn, but we could also say it's clockwise because it's a -270° turn. The conclusion seems then that the rotation has no direction in itself.