Of course there are results implying such things in degree theory and as @unknown (google) mentions.

Our "teaching" proof of Sperner's Lemma here http://link.springer.com/article/10.1007%2Fs00199-007-0257-0 develops this intuition for the purpose of pedagogy. So the connection between what you are talking about and Sperner's lemma is there and simply demonstrated (you'll have to do slight extra thinking in the middle of the homotopy that we construct).

In class every year I present this folding idea through a very nice story that I should have  written up but have not:

**Can we use maths to design an invisibility cloak for Harry Potter**?

Harry Potter has an invisibility cloak made of cloth having very simple properties:
It has two sides: Front and Back. We will write F|B and B|F to denote the orientation of the cloth.

   

 - if an observer looks that the Front side, then she does **not** see the first object behind the cloth and only sees the second object behind it.
 - if an observer looks at the back side, then she **will** see the first object behind the cloth (i.e., looking at it from Back side it behaves like glass). 


**A.** So consider this

                     Severus-Snape       F|B Harry B|F      Draco-Malfoy

Professor Snape sees Draco and Draco sees Snape. Harry is invisible.

**B.** But consider this material with one fold

                     Severus-Snape       F|B Harry B|F B|F      Draco-Malfoy

Draco-Malfoy sees Harry and Severus-Snape sees Draco. Bloody-hell as Ron would say, not only is Harry visible but Snape and Malfoy have figured out that he has an invisibility cloak on.  

**C.** Similarly consider this material with one fold

                     Severus-Snape       F|B Harry B|F F|B      Draco-Malfoy

Draco-Malfoy sees Severus-Snape but Severus-Snape does not see Draco. Snape and Malfoy have figured out that an invisibility cloak is being used. 


So how can we design an invisibility cloak that makes sure that Harry is not found out even if he runs. Use degree theory of course (and thus fixed point theory is important).  

Put the cloth around Harry from head to toe as follows

F|B Harry B|F

make sure there are no folds initially. Sew the cloak so that there are no holes in it whatsoever.

If the cloth is folding without ripping (i.e., Harry uses it in the way it is designed) all folds will be of the following form 

              Harry B|F F|B B|F     Draco-Malfoy 
 
Odd numbered and if we let B|F be of orientation +1 and denote by -1 the orientation F|B we see any old configuration will have +1 as the sum of the orientations of the folds. In particular, Harry is never found out.

A clever student always asks about the seam line of the folds. Well by Sard's theorem you cannot see those.