We revisit duality-covariant higher-derivative corrections which arise from the generalized Bergshoeff-de Roo (gBdR) identification, a prescription that gives rise to a two parameter family of $α'$-corrections to the low-energy effective action of the bosonic and the heterotic string. Although it is able to reproduce all corrections at the leading and sub-leading ($α'^2$) order purely from symmetry considerations, a geometric interpretation, like for the two-derivative action and its gauge transformation is lacking. To address this issue and to pave the way for the future exploration of higher-derivative (=higher-loop for the $β$-functions of the underlying $σ$-model) corrections to generalized dualities, consistent truncations and integrable $σ$-models, we recover the gBdR identification's results from the \PS{} construction that provides a natural notion of torsion and curvature in generalized geometry.