T-duality building blocks for?0 string corrections

T-duality has been shown to constrain the higher-derivative corrections of string theory. We revisit the problem of understanding the T-duality constraints imposed on the alpha 0 corrections using the language of a torsionful connection. We find a convenient way to express the Ood; d thorn invariants in terms of linear combinations of the metric and B-field for general d-dimensional torus compactifications, which we then use to revisit the heterotic and bosonic ten-dimensional string action at order alpha 0. We also comment on the four-point functions for corrections of order alpha 03 in the type II string.