  We discover an Ising-type duality in the general $N$-state chiral Potts
model, which is the Kramers-Wannier duality of planar Ising model when N=2.
This duality relates the spectrum and eigenvectors of one chiral Potts model at
a low temperature (of small $k'$) to those of another chiral Potts model at a
high temperature (of $k'^{-1}$). The $\tau^{(2)}$-model and chiral Potts model
on the dual lattice are established alongside the dual chiral Potts models.
With the aid of this duality relation, we exact a precise relationship between
the Onsager-algebra symmetry of a homogeneous superintegrable chiral Potts
model and the $sl_2$-loop-algebra symmetry of its associated
spin-$\frac{N-1}{2}$ XXZ chain through the identification of their eigenstates.
