  We investigate $\alpha'$ corrections of bosonic strings in the framework of
double field theory. The previously introduced "doubled $\alpha'$-geometry"
gives $\alpha'$-deformed gauge transformations arising in the Green-Schwarz
anomaly cancellation mechanism but does not apply to bosonic strings. These
require a different deformation of the duality-covariantized Courant bracket
which governs the gauge structure. This is revealed by examining the $\alpha'$
corrections in the gauge algebra of closed string field theory. We construct a
four-derivative cubic double field theory action invariant under the deformed
gauge transformations, giving a first glimpse of the gauge principle underlying
bosonic string $\alpha'$ corrections. The usual metric and $b$-field are
related to the duality covariant fields by non-covariant field redefinitions.
