  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 $\alpha'$ corrections using the language of a
torsionful connection. We find a convenient way to express the $O(d,d)$
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 $\alpha'$. We
also comment on the four-point functions for corrections of order $\alpha'^3$
in the type II string.
