This theoretical work demonstrates the existence of seven new shear-horizontal surface acoustic waves (SH-SAWs) when different electrical and magnetic boundary conditions are applied. All the discovered SH-SAWs can propagate on the surface of the cubic piezoelectromagnetics along direction [101]. Calculations of the new SH-SAW features were performed for two-phase composites with the piezomagnetic phase (Metglas, Terfenol-D, Galfenol, Alfenol) and the piezoelectric phase (Tl3TaSe4, Tl3VS4, PZT). It was also found that the surface Bleustein-Gulyaev-Melkumyan wave can propagate in the cubic...