TITLE: Klein bottle as two Möbius strips.
QUESTION [20 upvotes]: 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?

REPLY [3 votes]: 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.