  Generalised hydrodynamics predicts universal ballistic transport in
integrable lattice systems when prepared in generic inhomogeneous initial
states. However, the ballistic contribution to transport can vanish in systems
with additional discrete symmetries. Here we perform large scale numerical
simulations of spin dynamics in the anisotropic Heisenberg $XXZ$ spin $1/2$
chain starting from an inhomogeneous mixed initial state which is symmetric
with respect to a combination of spin-reversal and spatial reflection. In the
isotropic and easy-axis regimes we find non-ballistic spin transport which we
analyse in detail in terms of scaling exponents of the transported
magnetisation and scaling profiles of the spin density. While in the easy-axis
regime we find accurate evidence of normal diffusion, the spin transport in the
isotropic case is clearly super-diffusive, with the scaling exponent very close
to $2/3$, but with universal scaling dynamics which obeys the diffusion
equation in nonlinearly scaled time.
