We present high-precision optical polarimetric observations of the black hole X-ray binary Cygnus X-1 that span several cycles of its 5.6-day orbital period. The week-long observations on two telescopes located in opposite hemispheres allowed us to track the evolution of the polarization within one orbital cycle with the highest temporal resolution to date. Using the field stars, we determined the interstellar polarization in the source direction and subsequently its intrinsic polarization P_int = 0.82% ± 0.15% with a polarization angle θ_int = 155° ± 5°. The optical polarization angle is aligned with that in the X-rays recently obtained with the Imaging X-ray Polarimetry Explorer. Furthermore, it is consistent within the uncertainties with the position angle of the radio ejections. We show that the intrinsic polarization degree is variable with the orbital period with an amplitude of ∼0.2% and discuss various sites of its production. Assuming that the polarization arises from a single Thomson scattering of the primary star radiation by the matter that follows the black hole in its orbital motion, we constrained the inclination of the binary orbit i > 120° and its eccentricity e < 0.08. The asymmetric shape of the orbital profiles of the Stokes parameters also implies the asymmetry of the scattering matter distribution in the orbital plane, which may arise from the tilted accretion disk. We compared our data to the polarimetric observations made in 1975–1987 and find good agrement within 1° between the intrinsic polarization angles. On the other hand, the polarization degree decreased by 0.4% over half a century, suggesting secular changes in the geometry of the accreting matter.