Electronic band-structure calculations on Pr3I3Ru and Y3I3Ru have been performed in order to analyze the large structural differences found in these isoelectronic compounds. These constitute two structural extremes within the family of monoclinic RE3I3Ru phases (RE = rare-earth metal) that exhibit distortions ranging from one-dimensional double chains of trans-edge-sharing octahedra (bioctahedral chains, BOH) to one-dimensional chains of trans-edge-sharing square pyramidal units bonded base to base (bisquare pyramidal chains, BSP). The structure of La3I3Ru was established by single-crystal X-ray diffraction (monoclinic, P2(1)/m, Z = 4, a = 9.343(1) Angstrom, b = 4.3469(8) Angstrom ,c = 12.196(3) Angstrom, beta = 93.42(2)degrees) and found to be isomorphous with the BOH Pr3I3Ru. It is determined thar the structural variation in this RE(3)I(3)Z family of materials depends largely on the differences in orbital energies between the corresponding rare-earth metal and the interstitial. These bonding considerations can be generalized to account for structural variations in a variety of other rare-earth halides as well as several group 4 or 5 reduced metal halide cluster phases.