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It is shown that an anisotropic composite structure, which consists of alternating layers with either negative permittivity or negative permeability, can switch between optical responses characteristic of right-handed and left-handed materials depending on the polarization (TE or TM) of the incident electromagnetic field. The switching possibility is maintained for a small angular range around normal incidence, the behavior of the structure for larger incidence angles showing superluminal group velocities characteristic for anisotropic metamaterials.