  We determine the spin-exchange dynamical structure factor of the Heisenberg
spin chain, as is measured by indirect Resonant Inelastic X-ray Scattering
(RIXS). We find that two-spin RIXS excitations nearly entirely fractionalize
into *two spinon* states. These share the same continuum lower bound as
single-spin neutron scattering excitations, even if the relevant final states
belong to orthogonal symmetry sectors. The RIXS spectral weight is mainly
carried by higher-energy excitations, and is beyond the reach of the low-energy
effective theories of Luttinger liquid type.
