Single-phase Ln(4)Ni(3)O(8) (Ln = La, Nd) nickelates were synthesized and their crystal structures were determined by Rietveld refinement of powder neutron diffraction data. The crystal structures of these mixed-valent Ni1+/Ni2+ phases belong to the T'-type and are built by intergrowth of LnO(2) fluorite layers with triple NiO2 infinite-layer structural blocks. The major driving force of transformation of the LnO rock-salt block of the parent Ln(4)Ni(3)O(10-delta) Ruddlesden-Popper phases to the fluorite arrangement in the reduced Ln(4)Ni(3)O(8) phases is attributed to internal structural stress. This transformation allows longer Ni-O bonds in Ln(4)Ni(3)O(8) without overstretching of the Ln-O bonds, especially in the equatorial plane. The observed displacement of Ni atoms from the outer NiO2 planes toward the Ni atom of the central NiO2 plane in Ln(4)Ni(3)O(8) is ascribed to large electrostatic repulsion from the fluorite part of the structure. X-ray absorption spectra near the K-edge of Ni suggest that the charge density on the nickel ion is similar for all members of the T'-type Ln+1NinO2n+2 homologous series, which correlates with nearly constant Ni-O bond lengths observed in all the reduced nickelates. This suggests that the formal changes in the valence state of Ni affect the covalency of the Ni-O bond.