Connected sum of projective plane $\cong$ Klein bottle How can I see that the connected sum $\mathbb{P}^2 \# \mathbb{P}^2$ of the projective plane is homeomorphic to the Klein bottle?
I'm not necessarily looking for an explicit homeomorphism, just an intuitive argument of why this is the case. Can we see it using fundamental polygons?

[Munkres]=Munkres J. R.: Elements of Algebraic Topology, Reading, MA: Addison-Wesley, 1984.
The proof is just a simple trick. Munkres, p.39, Figure 6.9 only gives one part of the proof. The orientation of each $2$-simplex in the figure is the counterclockwise direction; the left figure represents the Klein bottle. The other part of the proof is to establish a homeomorphism between the right figure in Munkres, p.39, Figure 6.9 and the first figure in Munkres, p.38, Figure 6.8 using the last two figures in Munkres, p.38, Figure 6.8. The last two figures in Munkres, p.38, Figure 6.8 represent $P^2\#P^2$. Let us consider its left branch first. It is a $2$-simplex denoted by the path $D\to D\to -C$ using its directed boundary segments rather than the order of its vertices. The minus sign represents the clockwise direction. The starting point and the end point of the path $D\to D$ are the same. $C$ is a closed curve. The same $2$-simplex can also be represented by $D\to -C\to D$. The corresponding $2$-simplex in the right figure of Munkres, p.39, Figure 6.9 is $C\to -B\to C$. Although the starting point and the end point of the path seem to look different, we can relabel vertices to make the path closed so that it may represent a $2$-simplex.
