Consider an open two dimensional manifold $M$, with its nontrivial boundary $\partial M$. Let $R$ be an orientation reversal transformation operator which flips the orientation of $M$ while keeps its boundary $\partial M$ fixed. For example, if $M=D^2$ is the north hemisphere with its boundary being the equator, then $R(M)$ is the south hemisphere and its boundary is still the equator. Then we can glue $M$ and $R(M)$ into a whole sphere $$D^2 \cup_{S^1} R(D^2) = S^2$$
As another example, let $M=\mathrm{Mobius}$ be a Mobius bend. $\partial \mathrm{(Mobius)}= S^1$. Then R(\mathrm{Mobius})=\mathrm{Mobius}, and $$\mathrm{Mobius} \cup_{S^1} R(\mathrm{Mobius}) = KB$$ where $KB$ is a Klein bottle.
My question is whether there exists a two dimensional open manifold $M$, such that $$M \cup_{\partial M} R(M) = \mathbf{RP}^2$$
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A more general question is whether there exist other two dimensional non-orientable manifolds $\Sigma$ (apart from $KB$) such that the decomposition below happens? $$M \cup_{\partial M} R(M) = \Sigma$$