We investigate the “27-day” spectral component of different global (aa and Dst) and local (ζ) geomagnetic indices in two period ranges relevant to the Sun's synodic rotation, as manifested by magnetic activity: 24-28 days (short) and 28-32 days (long). Cross-correlation analysis of the respective energies of the short and long periods of the 27-day rotation signal in the same geomagnetic index reveals well-defined de-synchronization events during certain solar cycles. The largest de-synchronization in the past century occurred during solar cycle 21. De-synchronization events first occur in indices of the Dst (and ζ) family, and then in indices of the aa family. We found no evidence that the strength of the de-synchronization of the solar rotation signal in the ζ-index would depend on geomagnetic latitude. Applying the same analysis to proper solar indices (sunspot number, F10.7 radio flux, interplanetary magnetic field (IMF) series, solar wind speed), we find that only the Bz component of the IMF demonstrates a de-synchronization, during solar cycle 21, between the energies of the 27-day solar rotation signal in the short and long period ranges. We discuss possible implications of these results with respect to the evolution of the toroidal and poloidal components of the Sun's magnetic field and to its large-scale structures.