Heterotic/heterotic duality in D=6,4

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.