We consider the numerical solution of anisotropic diffusion equations on arbitrary polygonal grids using vertex-centered finite volume schemes. A symmetric linear scheme is suggested and constructed on primary and dual meshes using diffusion coefficients defined at cell centers. This symmetric scheme employs only vertex-centered unknowns and leads to symmetric positive definite systems. Its coercivity, stability and H1 error estimate are obtained theoretically on meshes with star-shaped cells. Numerical experiments demonstrate the second-order accuracy of the solution for heterogeneous and anisotropic problems on severely distorted grids. Moreover, the proposed scheme does not have the so-called numerical heat barrier issue suffered by most existing cell-centered and hybrid schemes.