OK, this is weird, I have to reorient when I close either eye. And I can make it work for either eye depending on what thumb I hold up. If I use my right thumb I'm right eyed, but if I hold up my left thumb then I'm left eyed. (or maybe I'm doing it wrong).

I just went back to the how to do this post, and when I hold my thumb out and focus on my thumb, I see two background objects, when I focus on the background object I see two thumbs. So I just have to pick one. Maybe I'm still doing it wrong, or maybe neither one is dominant. I don't know.