The isostructural phases BaMg4.787(3)In3.213 and SrMg4.837(2)In3.163 were synthesized via high-temperature reactions of the elements in welded Ta tubes. The orthorhombic crystal structure established for these by single-crystal X-ray diffraction means (space group Pnma, Z = 4) features a 3D Mg-In network formed from condensed 21-vertex polyhedra that are centered by Ba (Sr), viz., Ae@Mg5M7In9 (M = Mg/In). Three of eight independent network sites are co-occupied by > 90% Mg and < 10% In(M). The network structure is the inverse of that found in the isopointal Ca(CaIn4)Au-3, as clearly delineated by site charge estimates for the equivalent In-3 versus Au-3 positions. Tight-binding LMTO and extended HUckel calculations on the ordered AeMg(5)In(3) support the differentiation of Mg and Ae in the structure, with substantial participation of Mg in dominant Mg-In bonding, but mainly an electron donor role for Ae. The Fermi energies lie close to pseudo gaps but with substantial DOS remaining, consistent with their measured metallic properties. An unmixed ideal AeMg(5)In(3) would be isostructural with YCo5P3, with comparable polyhedra, but the small P in the latter lacks an effective interbridging role.