We study systematically the ≃ 0.1 − 1200 Hz quasi-periodic oscillations (QPOs) and broad noise components observed in the power spectra of non-pulsing neutron-star and black-hole low-mass X-ray binaries. We show that among these components we can identify two, occurring over a wide range of source types and luminosities, whose frequencies follow a tight correlation. The variability components involved in this correlation include neutron-star kilohertz QPOs and horizontal-branch oscillations, as well as black-hole QPOs and noise components. Our results suggest that the same types of variability may occur in both neutron-star and black-hole systems over three orders of magnitude in frequency and with coherences that vary widely but systematically. Confirmation of this hypothesis will strongly constrain theoretical models of these phenomena and provide additional clues to understanding their nature.