  In the XY regime of the XXZ Heisenberg model phase diagram, we demonstrate
that the origin of magnetically ordered phases is influenced by the presence of
solvable points with exact quantum coloring ground states featuring a
quantum-classical correspondence. Using exact diagonalization and density
matrix renormalization group calculations, for both the square and the
triangular lattice magnets, we show that the ordered physics of the solvable
points in the extreme XY regime, at $\frac{J_z}{J_\perp}=-1$ and
$\frac{J_z}{J_\perp}=-\frac{1}{2}$ respectively with $J_\perp > 0$,
adiabatically extends to the more isotropic regime $\frac{J_z}{J_\perp} \sim
1$. We highlight the projective structure of the coloring ground states to
compute the correlators in fixed magnetization sectors which enables an
understanding of the features in the static spin structure factors and
correlation ratios. These findings are contrasted with an anisotropic
generalization of the celebrated one-dimensional Majumdar-Ghosh model, which is
also found to be (ground state) solvable. For this model, both exact dimer and
three-coloring ground states exist at $\frac{J_z}{J_\perp}=-\frac{1}{2}$ but
only the two dimer ground states survive for any $\frac{J_z}{J_\perp} >
-\frac{1}{2}$.
