Bergman-type phases in the Na-Au-T (T = Ga, Ge, and Sn) systems were synthesized by solid-state means and structurally characterized by single-crystal X-ray diffraction studies. Two structurally related (1/1) Bergman phases were found in the Na-Au-Ga system: (a) a conventional Bergman-type (CB) structure, Na26AuxGa54-x, which features empty innermost icosahedra, as refined with x = 18.1 (3), Im (3) over bar, a = 14.512(2) angstrom, and Z = 2; (b) a stuffed Bergman-type (SB) structure, Na26AuyGa55-y, which contains Ga-centered innermost icosahedra, as refined with y = 36.0 (1), Im (3) over bar, a = 14.597(2) angstrom, and Z = 2. Although these two subtypes have considerable phase widths along with respective tie lines at Na approximate to 32.5 and 32.1 atom %, they do not merge into a continuous solid solution. Rather, a quasicrystalline phase close to the Au-poor CB phase and an orthorhombic derivative near the Au-rich SB phase lie between them. In contrast, only Au-rich SB phases exist in the Ge and Sn systems, in which the innermost icosahedra are centered by Au rather than Ge or Sn. These were refined for Na26Au40.93(5)Ge14.07(5) (Im (3) over bar, a = 14.581(2) angstrom, and Z = 2) and Na26Au39.83(6)Sn15.17(6) (Im (3) over bar, a = 15.009(2) angstrom, and Z = 2), respectively. Occupations of the centers of Bergman clusters are rare. Such centering and coloring correlate with the sizes of the neighboring icosahedra, the size ratios between electropositive and electronegative components, and the values of the average valence electron count per atom (e/a). Theoretical calculations revealed that all of these phases are Hume-Rothery phases, with evident pseudogaps in the density of states curves that arise from the interactions between Fermi surface and Brillouin zone boundaries corresponding to a strong diffraction intensity.