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We show that the yield ratio of free spectator neutrons produced in high-energy $^{96}$Zr+$^{96}$Zr to $^{96}$Ru+$^{96}$Ru collisions is a clean probe of the neutron-skin thickness of colliding nuclei and the slope parameter $L$ of the symmetry energy. The idea is demonstrated based on the proton and neutron density distributions via a state-of-the-art Skyrme-Hartree-Fock-Bogolyubov calculation. Among spectator nucleons given by the Glauber model, free spectator neutrons include those from direct production that survive from clusterization as well as those from deexcitation of heavy clusters described by the popular GEMINI model. More free neutrons are produced in collisions of $^{96}$Zr nucleus due to its larger neutron skin, compared to those produced in collisions of $^{96}$Ru nucleus with a smaller neutron skin. The difference of the free spectator neutron yield is further increased with the increasing difference of the neutron-skin thickness between $^{96}$Zr and $^{96}$Ru with a larger $L$ value, and the increase in ultracentral collisions is particularly insensitive to model details and experimental uncertainties. Since the production of free spectator neutrons is not affected by the complicated dynamics in the mid-rapidity region, the ratio of their multiplicities in ultracentral isobaric collisions is a robust observable for constraining the neutron skin and $L$ value.