Dependence of potential on only the *difference* between two variables

[AxiomOfChoice]

I'm reading a paper that considers a diatomic molecule living in two dimensions in which the central nucleus is fixed at the origin. Ignore the electrons for the time being. Let $(r_1,\theta_1)$ and $(r_2,\theta_2)$ describe the locations of the nuclei, and let the molecule be subject to a potential $V(r_1,r_2,\theta_1,\theta_2)$. The paper claims that, if the potential is the same when we rotate the entire molecule, we must have
$$V(r_1,r_2,\theta_1,\theta_2) = V'(r_1,r_2,\theta_2-\theta_1);$$
i.e., the potential only depends on the difference between $\theta_1$ and $\theta_2$. So the potential really only depends on three variables: $r_1$, $r_2$, and $\phi$, where $\phi = \theta_2 - \theta_1$. Can someone please explain why this is? I don't see it.

 

[nooma]

what are the angle representative of?