Gas-surface interactions which are described with the accommodation coefficients play an important role in the determination of the mass, momentum, and thermal energy transport in rarefied gas flows, aerosol mechanics, and vacuum technology. While the thermal accommodation coefficient has been extensively studied, the investigations of the other coefficients have been limited, partly because of the difficulties that are associated with the available theories and control of surfaces. To further understand the tangential momentum accommodation coefficients, experimental investigations of the rotational motion of a freely suspended sphere in a variety of gases and over a range of pressures were undertaken. It was found that the experimental results are generally well described by recent theoretical works that have provided numerical solutions of the linearized Boltzmann equation. Further, the theory can be used to interpret experimental data in terms of the tangential momentum accommodation coefficient, alpha over the entire range of the Knudsen number. The values of a are near 1 for most of the gas types and vary only slightly with the pressure.