I'm revising some string basics, and have come across the following problem. For closed strings one introduces the worldsheet parity operator
$$\Pi : \sigma \mapsto \ell-\sigma$$
where $\ell$ is the string length. In order to relate the left-movers and right-movers by parity in the Fourier expansion one assumes that $X$ and $P$ are parity even.
But why is this reasonable? I don't understand what forces $X$ and $P$ to have a specific behaviour under parity. Indeed surely a generic closed string will not have $X(\sigma)=X(\ell-\sigma)$!
Is there something I'm missing here? Thanks in advance for any helpful thoughts.
I wonder whether the notes I'm reading are only dealing with unoriented string theory. Perhaps that would be the reason for assuming the evenness of $X$ and $P$. Could someone possibly confirm that I've got the correct meaning of unoriented?