Well, it definitely sounds like it is out of kilter. Since it thinks it is flat when it is at an angle, then I'm guessing that is why it is going to landscape mode - it thinks you have it turned enough to want to go into that mode.
You can try doing a restore to Factory Settings, and then restore it to a backup (do an iTunes or iCloud backup before restoring):
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Or, since it is an easily reproducible problem, you can take it to the Apple Store. It should be obvious to anyone seeing your demo of the 0º at an angle issue that there is something that needs some sort of alignment....
Best of luck! 🙂
GB