We explore the cosmological implications of the clustering wedges, ξ⊥(s) and ξ∥(s), of the CMASS Data Release 9 sample of the Sloan Digital Sky Survey III (SDSS-III) Baryon Oscillation Spectroscopic Survey. These clustering wedges are defined by averaging the full two-dimensional correlation function, ξ(μ, s), over the ranges 0 < μ < 0.5 and 0.5 < μ < 1, respectively. These measurements allow us to constrain the parameter combinations DA(z)/rs(zd) = 9.03 ± 0.21 and cz/(rs(zd)H(z)) = 12.14 ± 0.43 at the mean redshift of the sample, z = 0.57. We combine the information from the clustering wedges with recent measurements of cosmic microwave background (CMB), baryon acoustic oscillations and Type Ia supernovae to obtain constraints on the cosmological parameters of the standard Λ cold dark matter (ΛCDM) model and a number of potential extensions. The information encoded in the clustering wedges is most useful when the dark energy equation of state is allowed to deviate from its standard ΛCDM value. The combination of all data sets shows no evidence of a deviation from a constant dark energy equation of state, in which case we find wDE = −1.013 ± 0.064, in complete agreement with a cosmological constant. We explore potential deviations from general relativity (GR) by constraining the growth rate ƒ(z) = d ln D(a)/d ln a, in which case the combination of the CMASS clustering wedges with CMB data implies ƒ(z = 0.57) = 0.719−0.096+0.092, in accordance with the predictions of GR. Our results clearly illustrate the additional constraining power of anisotropic clustering measurements with respect to that of angle-averaged quantities.