I am trying to identify the topological type of this fundamental polygon and I think it is the projective plane or the Klein bottle
If we treat the top green and red arrows a single arrow then we can call this $ab$
Doing a similar thing with the bottom two arrows we can combine them to get $a^{-1}b^{-1}$
Since they are inverses, can we swap the direction of $a^{-1}b^{-1}$ to get $ab$ in the opposite direction.... leading to the fundamental polygon of the projective plane?
I was thinking that other possibility is that it could be the Klein bottle since the new top and bottom arrows are in the same direction... but surely this would only work if $ab=a^{-1}b^{-1}$ ... i.e. the space is abelian... so maybe the Torus?!
I am really confused about this one so it would be good to gain some insight into what is happening, thanks