You're asking how a 2-D flatlander would represent a Möbius strip. The right analogy is not how we 3-D beings represent Klein bottles, but how we represent various compact 3-D spaces.
A useful method for representing these spaces is to start with some compact region, like a rectangle in 2-D or a cube in 3-D, and to glue some of the boundaries together. A Möbius strip is what you get when you glue one pair of opposite sides of a rectangle together with opposite orientations, as in the first picture below. The second picture represents a cube, with one pair of faces glued together with opposite orientations.
The words I used to describe these spaces, without any pictures, are perfectly good representations of the spaces, both for us and for flatlanders -- I could give a similar description in words (or equations) of a 4-D space defined by gluing specific boundaries of a hypercube together. I could tell you if it's orientable, what kind of loops exist in that that space, and many other things; but I have no way of drawing the space in all its 4-D glory.
For our 3-D space constructed by gluing opposite edges of a cube together, we have the picture above (which is a 2-D image of a transparent cube), and we could also take a block of wood and paint arrows on opposite faces to indicate the gluing. We would have to rotate the cube, or walk around it, to see the gluing pattern. This is how a flatlander would deal with our rectangle representation of the Möbius strip. A flatlander could also draw a 1-D image of the rectangle, by making the sides transparent. It wouldn't look like much to us, but it would make perfect sense to the flatlander.
Finally, a flatlander could paint a picture of what it would be like to live in a Möbius strip. Below is a picture of what it would look like to live in our 3-D space formed by gluing opposite sides of a cube: