We investigate charge pumping in carbon nanotube quantum dots driven by the electric field of a surface acoustic wave. We find that at small driving amplitudes, the pumped current reverses polarity as the conductance is tuned through a Coulomb blockade peak using a gate electrode. We study the behavior as a function of wave amplitude, frequency and direction and develop a model in which our results can be understood as resulting from adiabatic charge redistribution between the leads and quantum dots on the nanotube.