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I have been looking at the following fundamental polygon: $A^{-1} B A^{-1} B$.

After thinking about it I believe this should create a torus, is this right? And if not what shape does it create?

I suspect I am wrong because this article suggests that the only fundamental polygons that create a torus are: $A B A^{-1} B^{-1}$ and a hexagonal one.

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    Have you drawn the unit square with these sides? – 2017-01-08
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    @RahmaAlfarsy: I added a picture to explain the part about the projective plane. Maybe this was not obvious for you. I hope the answer is acceptable, now. – 2017-01-09
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    @DanielBernoulli Thank you the diagrams were very useful – 2017-01-19

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Indeed $ABA^{-1}B^{-1}$ represents the torus. As you can see in this animation.

$ABAB$ represents the projective plane. For further information visit Wikipedia. Here is a picture:

real projective plane

EDIT: Maybe it is not that obvious to you, but by cutting the square of the representation $A^{-1}BA^{-1}B$ along the diagonal line and by gluing things in another order, we get exactly the projective plane $ABAB$ (with the picture anove) or $C^2$ (as in the picture below).

real projective plane 2