We present and analyze a stabilized hybridized Nitsche method for elliptic problems with sign-changing coefficients without imposing symmetry assumptions on the mesh around the material interfaces. The use of a stabilized primal-dual formulation allows us to cope with the sign-changing nature of the problem and to prove optimal error estimates under two assumptions on the continuous problem, namely that it admits a unique solution and that the contrast at the sign-changing interface lies outside a certain critical interval. The method can be used on arbitrary shape-regular meshes (fitted to material interfaces) and yields optimal convergence rates for smooth solutions. As an illustration, the method is applied to simulate a realistic acoustic cloaking device.