We present a measurement of the anisotropic void-galaxy cross-correlation function in the CMASS galaxy sample of the BOSS DR12 data release. We perform a joint fit to the data for redshift space distortions (RSD) due to galaxy peculiar velocities and anisotropies due to the Alcock-Paczynski (AP) effect, for the first time using a velocity field reconstruction technique to remove the complicating effects of RSD in the void centre positions themselves. Fits to the void-galaxy function give a 1% measurement of the AP parameter combination DA(z)H(z)/c = 0.4367 ± 0.0045} at redshift z = 0.57, where DA is the angular diameter distance and H the Hubble parameter, exceeding the precision obtainable from baryon acoustic oscillations (BAO) by a factor of ∼3.5 and free of systematic errors. From voids alone we also obtain a 10%} measure of the growth rate, ƒσ8(z = 0.57) = 0.501 ± 0.051}. The parameter degeneracies are orthogonal to those obtained from galaxy clustering. Combining void information with that from BAO and galaxy RSD in the same CMASS sample, we measure DA(0.57)/rs = 9.383 ± 0.077 (at 0.8% precision), H(0.57)rs=(14.05±0.14)103 kms−1Mpc−1 (1%) and ƒσ8 = 0.453 ± 0.022 (4.9%)}, consistent with cosmic microwave background (CMB) measurements from Planck. These represent a factor ∼2 improvement in precision over previous results through the inclusion of void information. Fitting a flat cosmological constant ΛCDM model to these results in combination with Planck CMB data, we find up to an 11%} reduction in uncertainties on H0 and Ωm compared to use of the corresponding BOSS consensus values. Constraints on extended models with non-flat geometry and a dark energy of state that differs from w = −1 show an even greater improvement.