  We study the interaction of Rayleigh and shear horizontal surface acoustic
waves (SAWs) with spin waves in thin Ni films on a piezoelectric LiTaO$_3$
substrate, which supports both SAW modes simultaneously. Because Rayleigh and
shear horizontal modes induce different strain components in the Ni thin films,
the symmetries of the magnetoelastic driving fields, of the magnetoelastic
response, and of the transmission nonreciprocity differ for both SAW modes. Our
experimental findings are well explained by a theoretical model based on a
modified Landau--Lifshitz--Gilbert approach. We show that the symmetries of the
magnetoelastic response driven by Rayleigh- and shear horizontal SAWs
complement each other, which makes it possible to excite spin waves for any
relative orientation of magnetization and SAW propagation direction and,
moreover, can be utilized to characterize surface strain components of unknown
acoustic wave modes.
