We present the high-precision optical polarimetric observations of black hole X-ray binary Cyg X-1, spanning several cycles of its 5.6 day orbital period. 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 determine the interstellar polarization in the source direction and subsequently its intrinsic polarization. The optical polarization angle is aligned with that in the X-rays as 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 PD is variable with the orbital period with the amplitude of $\sim$0.2% and discuss various sites of its production. Assuming 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 constrain the inclination of the binary orbit $i>120^\circ$ and its eccentricity $e<0.08$. The asymmetric shape of the orbital profiles of Stokes parameters implies also the asymmetry of the scattering matter distribution about the orbital plane, which may arise from the tilted accretion disk. We compare our data to the polarimetric observations made over 1975-1987 and find good, within $1^\circ$, agreement between the intrinsic polarization angles. On the other hand, the PD decreased by 0.4% over half a century, suggesting the presence of secular changes in the geometry of accreting matter.