  Two symmetries of the local nonlinear delta-f gyrokinetic system of equations
in tokamaks in the high flow regime are presented. The turbulent transport of
toroidal angular momentum changes sign under an up-down reflection of the
tokamak and a sign change of both the rotation and the rotation shear. Thus,
the turbulent transport of toroidal angular momentum must vanish for up-down
symmetric tokamaks in the absence of both rotation and rotation shear. This has
important implications for the modeling of spontaneous rotation.
