Usually, the pop explanations of the properties of the Möbius strip say that if you take a pencil and start drawing a line in the middle of the strip, you'll have to make two complete loops to return to the original position. But if you look at the parametrization of the strip, after one loop you are again at the same position, albeit with the opposite chirality: it's like to say that the ink passes through the paper and colors both "faces" of the strip.
Which is the correct way to look at the strip?