Variable active galactic nuclei showing periodic light curves have been proposed as massive black hole binary (MBHB) candidates. In such scenarios, the periodicity can be due to relativistic Doppler-boosting of the emitted light. This hypothesis can be tested through the timing of scattered polarized light. Following the results of polarization studies in type I nuclei and of dynamical studies of MBHBs with circumbinary discs, we assume a coplanar equatorial scattering ring, whose elements contribute differently to the total polarized flux, due to different scattering angles, levels of Doppler boost, and line-of-sight time delays. We find that in the presence of an MBHB, both the degree of polarization and the polarization position angle have periodic modulations. The polarization angle oscillates around the semiminor axis of the projected MBHB orbital ellipse, with a frequency equal either to the binary's orbital frequency (for large scattering screen radii), or twice this value (for smaller scattering structures). These distinctive features can be used to probe the nature of periodic MBHB candidates and to compile catalogues of the most promising sub-pc MBHBs. The identification of such polarization features in gravitational-wave (GW) detected MBHBs would enormously increase the amount of physical information about the sources, allowing the measurement of the individual masses of the binary components, and the orientation of the line of nodes on the sky, even for monochromatic GW signals.