  We consider $E_8\times E_8$ heterotic compactifications on $K3$ and $K3\times
T^2$. The idea of heterotic/heterotic duality in $D=6$ has difficulties for
generic compactifications since for large dilaton values some gauge groups
acquire negative kinetic terms. Recently Duff, Minasian and Witten (DMW)
suggested a solution to this problem which only works if the compactification
is performed assuming the presence of symmetric gauge embeddings on both
$E_8$'s. We consider an alternative in which asymmetric embeddings are possible
and the wrong sign of kinetic terms for large dilaton value is a signal of
spontaneous symmetry breaking. Upon further toroidal compactification to $D=4$,
we find that the duals in the DMW case correspond to $N=2$ models in which the
$\beta$-function of the different group factors verify ${\beta }_\alpha=12$,
whereas the asymmetric solutions that we propose have ${\beta }_\alpha=24$. We
check the consistency of these dualities by studying the different large $T,S$
limits of the gauge kinetic function. Dual $N=1$, $D=4$ models can also be
obtained by the operation of appropriate freely acting twists, as shown in
specific examples.
