This paper solves the combinatorics relating the intersection theory of $ψ$-classes of Hassett spaces to that of $\overline{\mathcal{M}}_{g,n}$. A generating function for intersection numbers of $ψ$ classes on all Hassett spaces is obtained from the Gromov-Witten potential of a point via a non-invertible transformation of variables. When restricting to diagonal weights, the changes of variables are invertible and explicitly described as polynomial functions. Finally, the comparison of potentials is extended to the level of cycles: the pinwheel cycle potential, a generating function for tautological classes of rational tail type on $\overline{\mathcal{M}}_{g,n}$ is the right instrument to describe the pull-back to $\overline{\mathcal{M}}_{g,n}$ of all monomials of $ψ$ classes on Hassett spaces.