Heteroepitaxial non-polar III-Nitride layers may exhibit extensive anisotropy in the surface morphology and the epilayer microstructure along distinct in-plane directions. The structural anisotropy, evidenced by the “M”-shape dependence of the (112¯0) x-ray rocking curve widths on the beam azimuth angle, was studied by combining transmission electron microscopy observations, Raman spectroscopy, high resolution x-ray diffraction, and atomic force microscopy in a-plane GaN epilayers grown on r-plane sapphire substrates by plasma-assisted molecular beam epitaxy (PAMBE). The structural anisotropic behavior was attributed quantitatively to the high dislocation densities, particularly the Frank-Shockley partial dislocations that delimit the I1 intrinsic basal stacking faults, and to the concomitant plastic strain relaxation. On the other hand, isotropic samples exhibited lower dislocation densities and a biaxial residual stress state. For PAMBE growth, the anisotropy was correlated to N-rich (or Ga-poor) conditions on the surface during growth, that result in formation of asymmetric a-plane GaN grains elongated along the c-axis. Such conditions enhance the anisotropy of gallium diffusion on the surface and reduce the GaN nucleation rate.