Energy levels splitting in double-well potentials

As it is well known, the tunnel splitting (which are certain differences between energy levels) is the characteristic of the energy spectrum for the double-well potentials. When we calculate the energy eigenvalues for the ground state and the other higher states, we see that the energy levels satisfy the inequality $E_1-E_0 < E_3-E_2 < E_5-E_4 < …$ which implies that the adjacent energy levels are paired together. For example, $E_1$ is close to $E_0$ but far from $E_2$. Similarly, $E_2$ is far from $E_1$ but close to $E_3$. My question is that why we observe such a manner in double-well potentials? In other words, what is the physical interpretation of this effect?