The statistics of velocity fluctuations of turbulent Taylor-Couette flow are examined. The rotation rates of the inner and outer cylinders are varied while keeping the Taylor number fixed to 1.49×10(12) [O(Re)=10(6)]. The azimuthal velocity component of the flow is measured using laser Doppler anemometry. For each experiment 5×10(6) data points are acquired and carefully analyzed. Using extended self-similarity [Benzi et al., Phys. Rev. E 48, R29 (1993)] the longitudinal structure function exponents are extracted and are found to weakly depend on the ratio of the rotation rates. For the case where only the inner cylinder rotates the results are in good agreement with results measured by Lewis and Swinney [Phys. Rev. E 59, 5457 (1999)] using hot-film anemometry. The power spectra show clear -5/3 scaling for the intermediate angular velocity ratios -ω(o)/ω(i)∈{0.6,0.8,1.0}, roughly -5/3 scaling for -ω(o)/ω(i)∈{0.2,0.3,0.4,2.0}, and no clear scaling law can be found for -ω(0)/ω(i)=0 (inner cylinder rotation only); the local scaling exponent of the spectra has a strong frequency dependence. We relate these observations to the shape of the probability density function of the azimuthal velocity and the presence of a neutral line.