On the basis of ab-initio total-energy electronic-structure calculations, we find that interface localized electron states at the SiC/SiO$_2$ interface emerge in the energy region between 0.3 eV below and 1.2 eV above the bulk conduction-band minimum (CBM) of SiC, being sensitive to the sequence of atomic bilayers in SiC near the interface. These new interface states unrecognized in the past are due to the peculiar characteristics of the CBM states which are distributed along the crystallographic channels. We also find that the electron doping modifies the energetics among the different stacking structures. Implication for performance of electron devices fabricated on different SiC surfaces are discussed.