Why It's said to be that when a Flatlander makes a turn around a Möbius Strip, their internal organs are reversed, while they turn upside down?

Question

I mostly hear that a flatlander becomes their mirror counterpart when they make a turn inside it. Though for that to happen, they need to be turned upside down. Does it not make a difference when they are turned upside down inside 2D world? If so, how?

Let's say there is a stick coming from below, the flatlander will see a stick coming from below at first, when they make a full turn inside the Möbius strip, they'll see it coming it from above. So how can we say that it does not make a difference?

Thank you in advance, this is my first question in Math Stackexchange so sorry if I made a mistake.

Answer 1

If you observe the front and the back of the creature and follow it like below then you will see that not only turns it up side down but its back gets to the front: