I have a 4 element 10 m yagi antenna. If I point it at the sky, will its azimuth radiation pattern then become the same as its elevation radiation pattern when horizontally polarized?
Short answer: yes, provided there are no ground effects in either orientation. That's how azimuth and elevation are defined.
If there are ground effects in either orientation, then it is likely that the radiation patterns will change as the orientation of the antenna, in relation to ground, is changed. In this case it is likely that changing the orientation of the antenna will not cause the azimuth pattern to become to the elevation pattern and visa versa.