Altermagnets (AMs) constitute a novel class of spin-compensated materials in which opposite-spin sublattices are connected by a crystal rotation, causing their electronic isoenergy surfaces to be spin split. While cubic and tetragonal crystal symmetries tend to produce AMs in which the splitting of electronic isoenergy surfaces has d-wave symmetry, hexagonal AMs, such as CrSb and MnTe, are g-wave AMs. Here we investigate the purely magnetic modes and spin textures of g-wave AMs and show that they are drastically different for easy-axial (CrSb) and easy-planar (MnTe) materials. We show that in CrSb the splitting of the chiral magnon branches possesses g-wave symmetry, with each branch carrying a fixed momentum-independent magnetic moment. The altermagnetic splitting is not affected by the easy-axial anisotropy and is the same as that in the nonrelativistic limit. The magnon splitting of MnTe, however, does not strictly possess g-wave symmetry due to its easy-planar anisotropy. Instead, the magnetic moment of each branch becomes momentum dependent, with a distribution that is of g-wave symmetry. To generalize the concept of the altermagnetic splitting beyond the nonrelativistic limit, we introduce an alternative, directly observable splitting parameter that encompasses both the magnon eigenenergy and its magnetic moment (a component along the ground-state Néel vector), and exhibits g-wave symmetry in both easy-axial and easy-planar cases. The associated altermagnetic domain walls in easy-axial CrSb possess a net magnetization with an amplitude that depends on their orientation.