  We study heterotic supergravity at $\mathcal{O}(\alpha')$, first described in
detail in 1989 by Bergshoeff and de Roo. In particular, we discuss an ambiguity
of a connection choice on the tangent bundle. It is well known that at
$\mathcal{O}(\alpha')$ the Hull connection gives a consistent supergravity
theory with supersymmetry transformations given in the usual way. We consider
deformations of this connection corresponding to field redefinitions, and the
necessary corrections to the supersymmetry transformations. We are interested
in the moduli space of such field redefinitions which allow for supersymmetric
solutions to the equations of motion. We show that for solutions on $M_4\times
X$, where $M_4$ is Minkowski and $X$ is compact, the moduli space of
infinitesimal field redefinitions is given by $H^{(0,1)}(X,\textrm{End}(TX))$.
This space corresponds to infinitesimally close connections for which the
equations of motion are satisfied. The setup suggests a symmetry between the
gauge connection and the tangent bundle connection, as also employed by
Bergshoeff and de Roo. We argue that this symmetry should be kept to higher
orders in $\alpha'$, and propose a natural choice for the corresponding tangent
bundle connection used in curvature computations. In particular, the Hull
connection should be corrected at second and higher orders in $\alpha'$ from
this point of view.
