I've been thinking about this for bit and wonder what others think.
Compositionally, it is generally considered better to give a subject "room to move" in the frame.
I've seen some comments on some photos that have a bird moving one way while looking to the rear that indicate there should be more room in the direction the bird is facing, rather than the way it is moving.
Personally, I get that "look out for the tree" feeling if there isn't room in the direction of motion rather than the direction the bird is looking.
Jim, I agree that there should be room left ahead of the direction of movement of the subject. If the head is turned toward us that is a plus. To my eye if the head is turned and there is no "room to move" then I get the impression the bird or animal is about to turn toward me.