We calculate the contribution from the two-photon exchange on the neutron to the hyperfine splitting of S energy levels. We update the value of the neutron Zemach radius and estimate total recoil and polarizability corrections. The resulting two-photon exchange in electronic atoms exceeds by an order of magnitude the leading Zemach term and has different sign both in electronic and muonic hydrogen.Modern spectroscopical measurements in light muonic atoms support the physics community with precise values of the Rydberg constant and nuclei electromagnetic radii [1][2][3]. The unexpected discrepancy between muonic and electronic values of the charge radius in hydrogen and deuterium [4-9] calls for revisiting the higher-order corrections with an emphasis on the uncertain hadronic and nuclei contributions. In particular, to analyze measurements of the hyperfine splitting in light muonic nuclei and to extract the precise value of the Zemach radius, the higher-order radiative corrections have to be taken into account [10,11]. In recent decades, the O α 5 contribution from the graph with two exchanged photons (TPE) on a proton and nucleus (see Fig. 1) to the Lamb shift and hyperfine splitting was scrutinized by numerous authors . Besides the scattering on a proton, the TPE effect in light atoms contains contributions from nuclei excitations as well as from the scattering on a neutron. The contribution from the two-photon exchange on the neutron to the Lamb shift was recently investigated in Ref. [39,40]. For the hyperfine splitting, only the leading Zemach correction was evaluated in Refs. [41,42] from parametrizations of the neutron form factors. FIG. 1: Two-photon exchange graph. The contribution of the crossed graph is included into the lower blob.In this work, we reproduce the result of Refs. [41,42] exploiting the modern form factor parametrizations which satisfy the consistency criterium of Ref. [34] for such a calculation. We account for the two-photon exchange effects beyond the Zemach term by means of the forward Compton scattering amplitudes [35]. We estimate the polarizability contribution from the MAID partial-wave solution and illustrate how good such an estimate can be on the example of the TPE on the pro-ton.We determine the hyperfine-splitting correction from the forward scattering amplitude at threshold. It is convenient to express the resulting hyperfine splitting and individual contributions in terms of the effective radii. The resulting nucleon radius r N 2γ is given as a sum of three terms:where r Z , r R and r pol stand for the Zemach, recoil and polarizability radii, respectively (the terminology is taken from the hydrogen TPE). These radii are expressed as the photon energy ν γ and virtuality Q 2 integrals over the neutron electric G E and magnetic G M form factors and polarized spin structure functions g 1 , g 2 [27,29,35,43,44]: