Summary: The relative ordering of energy levels is investigated for bound two-electron systems with potentials of the form \(V(r_1,r_2,r_{12})=Z(v(r_1)+v(r_2))-v(r_{12})\). Given the two one-body binding potentials \(v^{(1)}(r)\) and \(v^{(2)}(r)\), it is argued that if \(f(r)\equiv v^{(1)}(r)-v^{(2)}(r)\) is positive and monotonically decreasing upon increasing \(r\) then the corresponding eigenvalues of the two-electron Hamiltonians \[ \begin{aligned} \mathcal H_i & =-\frac{1}{2}(\nabla_1^2+\nabla_2^2)+Z(v^{(i)}(r_1)+v^{(i)}(r_2))-v^{(i)}(r_{12}),\\ i & = 1,2\end{aligned} \] are highly likely to be pairwise ordered, i.e., \(E_n^{(1)}\geq E_n^{(2)}\), \(n=1,2,\dots\), where \(E_1^{(i)}\leq E_2^{(i)}\leq\cdots\leq E_k^{(i)}\leq\cdots\), for both \(i=1\) and \(i=2\). This conjecture certainly holds at sufficiently large \(Z\). The range of \(n\) may be finite or infinite, depending on the nature of the potentials. In fact, the range of values of \(n\) for \(v^{(1)}(r)\) may be shorter than that of \(v^{(2)}(r)\) (which is more binding). The one-electron potentials specifically considered are: \[ v(r)=\begin{cases} v_C(r)=-\frac{1}{r}\quad & \text{Coulomb} \\ v_D(r)=-\frac{1}{r}\exp(-\lambda r)\quad & \text{Debye} \\ \quad & \text{(Yukawa)} \\ v_{H u}(r)=-\frac{\lambda}{\exp(\lambda r)-1}\quad & \text{Hulthén} \\ v_{ECSC}(r)=-\frac{1}{r}\exp(-\lambda r)\cos(\lambda r)\quad & \text{ECSC}^\ast,\end{cases}^\ast\text{Exponential-Cosine-Screened-Coulomb} \] where \(\lambda\) is the screening parameter. For each of the \(\lambda\)-dependent potentials we compare one- and two-electron spectra corresponding to distinct values of \(\lambda\). This is followed by pairwise comparison of distinct potentials.