  We construct an $O(d,d)$ invariant universal formulation of the first-order
$\alpha'$-corrections of the string effective actions involving the dilaton,
metric and two-form fields. Two free parameters interpolate between
four-derivative terms that are even and odd with respect to a $Z_2$-parity
transformation that changes the sign of the two-form field. The $Z_2$-symmetric
model reproduces the closed bosonic string, and the heterotic string effective
action is obtained through a $Z_2$-parity-breaking choice of parameters. The
theory is an extension of the generalized frame formulation of Double Field
Theory, in which the gauge transformations are deformed by a first-order
generalized Green-Schwarz transformation. This deformation defines a duality
covariant gauge principle that requires and fixes the four-derivative terms. We
discuss the $O(d,d)$ structure of the theory and the (non-)covariance of the
required field redefinitions.
