  We have investigated the electronic structure of Na$_x$Ca$_{1-x}$Cr$_2$O$_4$
using x-ray absorption spectroscopy together with Anderson impurity model
calculations with full multiplets. We show Na$_x$Ca$_{1-x}$Cr$_2$O$_4$ taking a
novel mixed-valence electronic state in which the positive charge-transfer (CT)
and the negative self-doped states coexist. While CaCr$_2$O$_4$ (one end
member) exhibits a typical CT nature with strong covalent character, Na
substitution causes a self-doped state with an oxygen hole. In NaCr$_2$O$_4$
(the other end member), positive CT and negative self-doped states coexist with
equal weight. This unusual electronic state is in sharp contrast to the
conventional mixed-valence description, in which the ground state can be
described by the mixture of Cr$^{3+}$ ($3d^3$) and Cr$^{4+}$ ($3d^2$).
