How Can We Understand Reflective Symmetry in a Square?

[srfriggen]

In my abstract algebra course we learned recently of the symmetries D4. Regarding flips/reflections, of which there are 4, it seems for the 2D object that is a square, you would have to "fold it through the 3rd dimension" to obtain a flip/reflection.

Couldn't you just invert the square by pulling the lines through each other, kind of like laying down a rubber band and pulling it through itself to obtain the desired effect? Of course assuming you can bend and stretch the lines of a square, and that they are able to pass through one another. I've seen something similar done to a sphere in a youtube video called, "How to turn a sphere inside out". Wondering if this is a feasible way to think of a reflection.

 

[Muphrid]

I think it's much clearer if you don't imagine the points moving through space. Rather, the transformation acts on the original square and produces some resulting image. It's a before and after sort of thing. There is no intermediate or in-between.