Fundamental properties of the planet Venus, such as its internal mass distribution and variations in length of day, have remained unknown. We used Earth-based observations of radar speckles tied to the rotation of Venus obtained in 2006-2020 to measure its spin axis orientation, spin precession rate, moment of inertia and length-of-day variations. Venus is tilted by 2.6392 ± 0.0008 deg (1σ) with respect to its orbital plane. The spin axis precesses at a rate of 44.58 ± 3.3 arcsec per year (1σ), which gives a normalized moment of inertia of 0.337 ± 0.024 and yields a rough estimate of the size of the core. The average sidereal day on Venus in the 2006-2020 interval is 243.0226 ± 0.0013 Earth days (1σ). The spin period of the solid planet exhibits variations of 61 ppm (~20 min) with a possible diurnal or semidiurnal forcing. The length-of-day variations imply that changes in atmospheric angular momentum of at least ~4% are transferred to the solid planet.