In the (MM'O-6)(infinity) chains of the transition-metal magnetic oxides Ca3MM'O-6 the MO6 trigonal prisms alternate with the M'O-6 octahedra by sharing their triangular faces. In the (Co2O6)(infinity) chains of Ca3Co2O6 (M = M' = Co) the spins are coupled ferromagnetically, but in the (FeRhO6)(infinity) chains of Ca3FeRhO6 (M = Fe, M' = Rh) they are coupled antiferromagnetically. The origin of this difference was probed by carrying out spin-polarized density functional theory electronic band structure calculations for ordered spin states of Ca3Co2O6 and Ca3FeRhO6. The spin state of a (MM'O-6)(infinity) chain determines the occurrence of direct metal-metal bonding between the adjacent trigonal prism and octahedral site transition-metal atoms. The extent of direct metal-metal bonding in the (Co2O6)(infinity) chains of Ca3Co2O6 is stronger in the intrachain ferromagnetic state than in the intrachain antiferromagnetic state, so that the intrachain ferromagnetic state becomes more stable than the intrachain antiferromagnetic state. Such a metal-metal-bonding-induced ferromagnetism is expected to occur in magnetic insulators and magnetic metals of transition-metal elements in which direct metal-metal bonding can be enhanced by ferromagnetic ordering. In the (FeRhO6)chains of Ca3FeRhO6 the ferromagnetic coupling does not lead to a strong metal-metal bonding and the adjacent spins interact by the Fe-O center dot center dot center dot O-Fe super-superexchange, hence leading to an antiferromagnetic coupling.