  Let $F$ be a homogeneous $(\alpha_1,\alpha_2)$ metric on the reductive
homogeneous manifold $G/H$. Firstly, we characterize the natural reductiveness
of $F$ as a local $f$-product between naturally reductive Riemannian metrics.
Secondly, we prove the equivalence among several properties of $F$ for its mean
Berwald curvature and S-curvature. Finally, we find an explicit flag curvature
formula when $F$ is naturally reductive.
