Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

algebraic-geometry differential-topology schemes vector-bundles coherent-sheaves

Leaving a community wiki answer to remove this from the unanswered list (once someone upvotes it): Yes, this is true. Georges Elencwacj edited his answer to another question in order to address this one.

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