So, I've been thinking about this on and off for the last couple hours. If we skip a lot of the fiber bundle formalism, then we can think of the cylinder as the product $[0, 2\pi) \times (-1, 1)$. For the Möbius strip, I think you have to do something like taking the unit square and identifying $(x, 0)$ with $(1 - x, 1)$.
For the "oriented lines + circle", we decided that we can roughly think of this as $[0, 2\pi) \times \mathbb R$, so it's like an infinitely long cylinder. So naturally, one wonders about an infinitely long Möbius strip, but I can't see that in my mind and I don't know if …