  The exact joint multifractal distribution for the scaling and winding of the
electrostatic potential lines near any conformally invariant scaling curve is
derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff
dimension of the points where the potential scales with distance $r$ as $H \sim
r^{\alpha}$ while the curve logarithmically spirals with a rotation angle
phi=lambda ln r. It obeys the scaling law f(\alpha,\lambda)=(1+\lambda^2)
f(\bar \alpha)-b\lambda^2 with \bar \alpha=\alpha/(1+\lambda^2) and
b=(25-c)/{12}$, and where f(\alpha)\equiv f(\alpha,0) is the pure harmonic
measure spectrum, and c the conformal central charge. The results apply to O(N)
and Potts models, as well as to {\rm SLE}_{\kappa}.
