Evidence for Heterotic/Heterotic Duality

We re-examine the question of heterotic - heterotic string duality in six dimensions and argue that the $E_8\times E_8$ heterotic string, compactified on $K3$ with equal instanton numbers in the two $E_8$'s, has a self-duality that inverts the coupling, dualizes the antisymmetric tensor, acts non-trivially on the hypermultiplets, and exchanges gauge fields that can be seen in perturbation theory with gauge fields of a non-perturbative origin. The special role of the symmetric embedding of the anomaly in the two $E_8$'s can be seen from field theory considerations or from an eleven-dimensional point of view. The duality can be deduced by looking in two different ways at eleven-dimensional $M$-theory compactified on $K3\times {\bf S}^1/\Z_2$.