Using first-principles calculations, we demonstrate that the magnetic exchange interaction and the magnetocrystalline anisotropy of biatomic Fe chains grown in the trenches of the (5x1) reconstructed Ir(001) surface depend sensitively on the atomic arrangement of the Fe atoms. Two structural configurations have been considered which are suggested from recent experiments. They differ by the local symmetry and the spacing between the two strands of the biatomic Fe chain. Since both configurations are very close in total energy they may coexist in experiment. We have investigated collinear ferro- and antiferromagnetic solutions as well as a collinear state with two moments in one direction and one in the opposite direction (up arrow down arrow up arrow-state). For the structure with a small interchain spacing, there is a strong exchange interaction between the strands and the ferromagnetic state is energetically favorable. In the structure with larger spacing, the two strands are magnetically nearly decoupled and exhibit antiferromagnetic order along the chain. In both cases, due to hybridization with the Ir substrate the exchange interaction along the chain axis is relatively small compared to free-standing biatomic iron chains. The easy magnetization axis of the Fe chains also switches with the structural configuration and is out-of-plane for the ferromagnetic chains with small spacing and along the chain axis for the antiferromagnetic chains with large spacing between the two strands. Calculated scanning tunneling microscopy images and spectra suggest the possibility to experimentally distinguish between the two structural and magnetic configurations.