  Despite nearly a century of study of the $S=1/2$ Heisenberg model on the
square lattice, there is still disagreement on the nature of its high-energy
excitations. By tuning toward the Heisenberg model from the exactly soluble
Ising limit, we find that the strongly attractive magnon interactions of the
latter naturally account for a number of spectral features of the Heisenberg
model. This claim is backed up both numerically and analytically. Using the
density matrix renormalization group method, we obtain the dynamical structure
factor for a cylindrical geometry, allowing us to continuously connect both
limits. Remarkably, a semi-quantitative description of certain observed
features arises already at the lowest non-trivial order in perturbation theory
around the Ising limit. Moreover, our analysis uncovers that high-energy
magnons are localized on a single sublattice, which is related to the
entanglement properties of the ground state.
