There's got to be a better way, but I figured you can achieve this by rotating the viewport. The way I managed to do that was by creating a BoxContainer node with a Viewport Container inside, the child of which is a Viewport node. The tree looks like this:
I just created a script and attached it to the BoxContainer and using the set_rotation() method, I was able to change the orientation of the screen.
As far as I know, the root of any scene is not the top node, but a viewport. Maybe you could rotate that root vieport directly, but I do not know how. Maybe something that has to do with canvas layers or so. That's why I am not adding this as an answer. If my solution covers you, though, let me know and I will turn this comment to an answer so that others can find it.