Millimeter wave communication has been a focal point of research currently and become more and more attractive. A dual-band left-handed material structure is proposed for the application of millimeter wave communication. The structure cell is composed of two S-shaped conductor planes with upper and lower asymmetric rings, which are printed reverse of each other on both sides of the dielectric substrate. Through choosing and optimizing the structural dimensions reasonably, the S parameter values with two transmission peaks at about 50 GHz and 60 GHz are obtained. The corresponding electromagnetic parameters are calculated by the S parameters inversion algorithm. Simulated results demonstrate that permittivity and permeability are simultaneously negative from 48.2~52.2 GHz and 58.7~60.8 GHz where the typical left-handed characteristic is shown. The research has certain reference significance for the design of multi-band left-handed material and millimeter wave communication application.