I had a homework problem that asked "Can you cut a Klein bottle into two Mobius bands?" which I carelessly read initially as "Can you cut a Mobius band into two Mobius bands? After realizing my mistake, I quickly demonstrated that the actual problem of cutting a Klein bottle into two Mobius bands, but after spending an hour or so attempting to cut a Mobius band into two Mobius bands I am curious if it is possible... My intuition is no, but I am unable to prove this rigorously. The best I could do was cutting a Mobius band into a Mobius band and a cylinder with two twists (still a cylinder though).
So does there exists a cutset $C$ of the Mobius band $M$ such that $C$ is a simple closed curve in $M$ and $M \smallsetminus C$ is the union of two disjoint open sets such that the closure of each in $M$ is homeomorphic to $M$?