Altermagnets have attracted intense interest because they have the advantages of both ferromagnets and antiferromagnets. However, their experimental identification remains challenging, in particular for the A-type altermagnets that account for a large group of material candidates. Here, we discover a kind of anisotropic surface spin waves in A-type altermagnets, which is absent in ferromagnets and conventional antiferromagnets. The anisotropic properties of these surface spin waves originate from a nontrivial altermagnetic spin-group symmetry, where spin-opposite layers are connected by a combined spin-space and real-space operation rather than a pure translation or inversion. This symmetry lifts the $𝒫𝒯$-protected degeneracy between opposite-chirality magnon modes, leading to chirality-dependent top-bottom localization and chirality-split equal-frequency contours. We further show that these two properties can be measured experimentally from the stray field and the resonance absorption spectrum, respectively. Our results provide a signature for detecting altermagnets and will inspire spin-based logic and information-storage devices.