We report a comparative study on the crystal structure and physical properties of two new high-entropy alloys (HEAs), Hf10Mo25W10Re35Ru20 and Ta10Mo25W10Re35Ru20. Despite a 10% difference in chemical composition, the former crystallizes in the noncentrosymmetric cubic α-Mn structure while the latter adopts the centrosymmetric tetragonal σ-type structure. For both HEAs, bulk superconductivity is discovered and their specific heat jumps can be well described by a modified BCS model, consistent with a fully gapped state. In the normal state, the resistivity exhibits a temperature-independent or weakly insulating behavior, consistent with the presence of strong disorder scattering. Through comparison with isostructural HEA superconductors, we show that the selection between the α-Mn and σ-type phases correlates with the valence electron concentration of HEAs and atomic size difference of constituent elements.