  We show that given a harmonic map $\varphi$ from a Riemann surface to a
classical compact simply connected inner symmetric space, there is a
$J_2$-holomorphic twistor lift of $\varphi$ (or its negative) if and only if it
is nilconformal. In the case of harmonic maps of finite uniton number, we give
algebraic formulae in terms of holomorphic data which describes their extended
solutions. In particular, this gives explicit formulae for the twistor lifts of
all harmonic maps of finite uniton number from a surface to the above symmetric
spaces.
