Novel surface integral equation (SIE) based theory of characteristic mode (TCM) formulations are proposed for homogeneous penetrable bodies. Analytical expressions for the characteristic eigenvalues are presented and good agreement between the numerical and analytical results is observed for both lossless and lossy, as well as for dielectric and magnetodielectric objects. The new formulations, without any postprocessing methods, avoid spurious modes and resonances appearing in existing SIE-based TCM formulations. For lossy objects the obtained eigenvalues are complex. The real part of an eigenvalue is related to the ratio of the reactive and radiated power, as in the original TCM formulation for PEC structures, and the imaginary part gives the ratio between the dissipated and radiated power.