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The identity of the mysterious “hidden-order” (HO) phase in the correlated f-electron superconductor (SC) URu₂Si₂ has eluded researchers for more than three decades. Substitution of transition metals into URu₂Si₂ at the Ru sites reveals how factors such as lattice parameters, charge carrier concentration, disorder and d-f electron hybridization influence the HO phase and SC. We discuss the effects of the co-substitution of equal amounts of Re and Rh, which are isoelectronic on average, into URu₂Si₂ to form URu_2-2xRe_xRh_xSi_2, pseudo ternary alloys based on the physical properties including structure and lattice parameters, electrical resistivity, magnetic susceptibility, and specific heat. The features in the physical properties that characterize the energy gap associated with the HO phase (e.g., exponential T-dependence of the specific heat) are suppressed rapidly with Re and Rh co-substitution level so that by x = 0.11, the HO phase transition is no longer discernible. The HO phase transition temperature (T_HO) and the gap decrease monotonically with Re-Rh concentration x.