Möbius squared, a better illustration or not?

Question

Making a Möbius sling from a slip of paper makes very different effects on the direction that gets the the short ends (North - South) and direction that get the long sides (East – West). The North – South loses its edges – they are glued together. If you travel long enough in either direction, you come back to the place where you began, but you also get the East – West direction shifted left to right. The East – West direction instead keeps the both edges. You can travel “to the end of the world” in East/westerly direction, the twist in the sling only makes impossible to say which is which. And if you pass the edge of that, you are getting nowhere! If you instead of cutting a slip of paper begin by cutting out a cross, and then makes the four legs of that into two Möbius slings, 90 degrees apart, wouldn’t the area where the crossed slings meet be a better illustration of the type of 2d surface that is called a Möbius band? Or at least easier to understand? You step outside the square in the middle, that is a picture of the 2d surface, continue following the strip and you end up on the other side, with right and left shifted? Or would the central meeting of the glued together Möbius slips be some other type of surface?