  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 $K3\times {\bf S}^1/\Z_2$.
