Question

Is there a rotation for which the orientation of the image is always the same as that of the preimage

Answer 1

Sure. 360 degrees, or any integer multiple thereof.

Of course, if it starts out as a circle, then ANY rotation qualifies.

Answer 2

Yes.

If the image has rotational symmetry of order N, there are N such rotatations.

Any image has rotational symmetry of at least order 1, which is to say it looks the same if you rotate it one full turn.