  We establish the sign of the linear magnetoelectric (ME) coefficient,
$\alpha$, in chromia, Cr$_2$O$_3$. Cr$_2$O$_3$ is the prototypical linear ME
material, in which an electric (magnetic) field induces a linearly proportional
magnetization (polarization), and a single magnetic domain can be selected by
annealing in combined magnetic (H) and electric (E) fields. Opposite
antiferromagnetic domains have opposite ME responses, and which
antiferromagnetic domain corresponds to which sign of response has previously
been unclear. We use density functional theory (DFT) to calculate the magnetic
response of a single antiferromagnetic domain of Cr$_2$O$_3$ to an applied
in-plane electric field at 0 K. We find that the domain with nearest neighbor
magnetic moments oriented away from (towards) each other has a negative
(positive) in-plane ME coefficient, $\alpha_{\perp}$, at 0 K. We show that this
sign is consistent with all other DFT calculations in the literature that
specified the domain orientation, independent of the choice of DFT code or
functional, the method used to apply the field, and whether the direct
(magnetic field) or inverse (electric field) ME response was calculated. Next,
we reanalyze our previously published spherical neutron polarimetry data to
determine the antiferromagnetic domain produced by annealing in combined E and
H fields oriented along the crystallographic symmetry axis at room temperature.
We find that the antiferromagnetic domain with nearest-neighbor magnetic
moments oriented away from (towards) each other is produced by annealing in
(anti-)parallel E and H fields, corresponding to a positive (negative) axial ME
coefficient, $\alpha_{\parallel}$, at room temperature. Since $\alpha_{\perp}$
at 0 K and $\alpha_{\parallel}$ at room temperature are known to be of opposite
sign, our computational and experimental results are consistent.
