Symmetry results for the area formula in homogeneous groups

We prove that if the shape of the metric unit ball in a homogeneous group enjoys a precise symmetry property, then the associated distance yields the standard form of the area formula. The result applies to some classes of smooth and nonsmooth submanifolds. We finally prove the equality between spherical measure and centered Hausdorff measure, under two different geometric conditions on the shape of the metric unit ball.