  We dimensionally reduce the spacetime action of bosonic string theory, and
that of the bosonic sector of heterotic string theory after truncating the
Yang-Mills gauge fields, on a $d$-dimensional torus including all
higher-derivative corrections to first order in $\alpha'$. A systematic
procedure is developed that brings this action into a minimal form in which all
fields except the metric carry only first order derivatives. This action is
shown to be invariant under ${\rm O}(d,d,\mathbb{R})$ transformations that
acquire $\alpha'$-corrections through a Green-Schwarz type mechanism. We prove
that, up to a global pre-factor, the first order $\alpha'$-corrections are
uniquely determined by ${\rm O}(d,d,\mathbb{R})$ invariance.
