Most existing cell-centered finite volume schemes need to introduce auxiliary unknowns in order to maintain the second-order accuracy when the mesh is distorted or the problem is discontinuous, so interpolation algorithms of auxiliary unknowns are required. Interpolation algorithms are not only difficult to construct, but also bring extra computation. In this paper, an interpolation-free cell-centered finite volume scheme is proposed for the heterogeneous and anisotropic convection-diffusion problems on arbitrary polyhedral meshes. Unlike most of the existing cell-centered schemes, this scheme has no auxiliary unknowns, so no interpolation algorithm is required. Most interestingly, the scheme can be adapted to any mesh topology and can handle any discontinuity strictly. Numerical experiments show that this new scheme is robust, possesses a small stencil, and has approximately second-order accuracy for both diffusion-dominated and convection-dominated problems.