Does 1-chloro-3-methylcyclobutane have symmetry

Yes.

Although cyclobutane is not perfectly flat, substituents of the four-membered ring can project above or below the average plane of the ring. Therefore, 1-chloro-3-methylcyclobutane exists as either a cis or a trans form. If both the methyl and chloro substituents are either above or below the ring, then the orientation is cis. If one of the substituents is above the ring and the other is below the ring, then the orientation is trans.