  We present a new class of dualities relating non-geometric Calabi-Yau
compactifications of type II string theory to T-fold compactifications of the
heterotic string, both preserving four-dimensional $\mathcal{N}=2$
supersymmetry. The non-geometric Calabi-Yau space is a $K3$ fibration over
$T^2$ with non-geometric monodromies in the duality group $O(\Gamma_{4,20})$;
this is dual to a heterotic reduction on a $T^4$ fibration over $T^2$ with the
$O(\Gamma_{4,20})$ monodromies now viewed as heterotic T-dualities. At a point
in moduli space which is a minimum of the scalar potential, the type II
compactification becomes an asymmetric Gepner model and the monodromies become
automorphisms involving mirror symmetries, while the heterotic dual is an
asymmetric toroidal orbifold. We generalise previous constructions to ones in
which the automorphisms are not of prime order. The type II construction is
perturbatively consistent, but the naive heterotic dual is not modular
invariant. Modular invariance on the heterotic side is achieved by including
twists in the circles dual to the winding numbers round the $T^2$, and this in
turn introduces non-perturbative phases depending on NS5-brane charge in the
type II construction.
