Several high-resolution Mössbauer spectra of yttrium iron garnet, Y3Fe5O12, have been fit as a function of temperature with a new model based on a detailed analysis of the spectral changes that result from a reduction from the cubic Ia3̅d space group to the trigonal R3̅ space group. These spectral fits indicate that the magnetic sextet arising from the 16a site in cubic symmetry is subdivided into three sextets arising from the 6f, the 3d, 3d, and the 1a, 1b, 2c sites in rhombohedral-axis trigonal symmetry. The 24d site in cubic Ia3̅d symmetry is subdivided into four sextets arising from four different 6f sites in R3̅ rhombohedral-axis trigonal symmetry, sites that differ only by the angles between the principal axis of the electric field gradient tensor and the magnetic hyperfine field assumed to be parallel with the magnetic easy axis. This analysis, when applied to the potential nuclear waste storage compounds Y(3-x)Ca(0.5x)Th(0.5x)Fe5O12 and Y(3-x)Ca(0.5x)Ce(0.5x)Fe5O12, indicates virtually no perturbation of the structural, electronic, and magnetic properties upon substitution of small amounts of calcium(II) and thorium(IV) or cerium(IV) onto the yttrium(III) 24c site as compared with Y3Fe5O12. The observed broadening of the four different 6f sites derived from the 24d site results from the substitution of yttrium(III) with calcium(II) and thorium(IV) or cerium(IV) cations on the next-nearest neighbor 24c site. In contrast, the same analysis applied to Y(2.8)Ce(0.2)Fe5O12 indicates a local perturbation of the magnetic exchange pathways as a result of the presence of cerium(IV) in the 24c next-nearest neighbor site of the iron(III) 24d site.