Klein bottle as two Möbius strips. I read that glueing together two Möbius strips along their edges creates a surface that is equivalent to the so-called Klein bottle.   
The Möbius strip comes in two versions that are mirrored versions of each other (wrt the chirality of the half-turn in the strip).
So glueing together two of them, does it matter whether they are mirrored versions of each other or not? Is there a different result?
 A: They are only in two version if you consider the Mobius strip as being an object embedded in three-dimensional ambient space. This becomes pretty moot when you consider the Klein bottle can't be embedded into $3$-space, so instead we usually only consider the Mobius band and the Klein bottle as topological spaces, in which case they don't come with 'chiralities' or indeed orientations in this case because they are non-orientable.
To put it succinctly: All Mobius bands are homeomorphic.
A: The paper On the number of Klein bottle types by Carlo H. Séquin (Journal of Mathematics and the Arts, Volume 7, Issue 2, 2013) provides some answers to the above issues. It is available online. 
In short: Mobius bands definitely come with two different handinesses,
and depending how you combine them, you can get different types of Klein bottles.
A: If you are careful with your deformations when you draw topological examples, you can prove it by making a very crude drawing, like the one below. The cut occurs at the vertical plane of symmetry of the bottle. It is fairly clear that the final objects are Mobius bands.


A: I quickly threw this animation together to demonstrate that the "figure-8" immersion of the Klein bottle admits a decomposition into two Möbius bands with the same "apparent handedness," whatever that means.

