Rotation is favorable for confinement, but a stellarator can rotate at high speeds if and only if it is sufficiently close to quasisymmetry. This article investigates how close it needs to be. For a magnetic field B = B0 + αB1, where B0 is quasisymmetric, αB1 is a deviation from quasisymmetry, and α ≪ 1, the stellarator can rotate at high velocities if α < 1/2, with the ion Larmor radius over the characteristic variation length of B0. The cases in which this result may break down are discussed. If the stellarator is sufficiently quasisymmetric in the above sense, the rotation profile, and equivalently, the long-wavelength radial electric field, are not set neoclassically; instead, they can be affected by turbulent transport. Their computation requires the O(2) pieces of both the turbulent and the long-wavelength components of the distribution function. This article contains the first step towards a formulation to calculate the rotation profile by providing the equations determining the long-wavelength components of the O(2) pieces.