<p>Duality covariant curvature and torsion tensors in double field theory/generalized geometry are central in analyzing consistent truncations, generalized dualities, and related integrable <i>σ</i>-models. They are constructed systematically with the help of a larger, auxiliary space in a procedure inspired by Cartan geometry originally proposed by Poláček and Siegel for bosonic strings. It pivots around a maximally isotropic group that captures the generalized structure group of the physical space. We show how dropping the isotropy condition on this group allows us to describe heterotic/type I strings. As an immediate application, we construct a new family of heterotic backgrounds that interpolates between the two-dimensional cigar and trumpet backgrounds.</p>