We calculate nuclear-polarization energy shifts in muonic 208Pb. We employ a relativistic field-theoretical calculation and evaluate the ladder, cross, and seagull terms of the two-photon exchange diagrams in both the Feynman and Coulomb gauges. Gauge independence is very well satisfied with the calculated nuclear-polarization energies. Using these results, we analyze fine-structure splitting energies of muonic 208Pb because of the presence of the persisting discrepancies between experiment and calculation. The present nuclear-polarization energies explain about half of the anomaly in the ∆2p fine-structure splitting energy, and only one-fourth of the anomaly in the ∆3p fine-structure splitting energy.