  We compute three-term semiclassical asymptotic expansions of counting
functions and Riesz-means of the eigenvalues of the Laplacian on spheres and
hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically
for Riesz-means we prove upper and lower bounds involving asymptotically sharp
shift terms, and we extend them to domains of $\mathbb S^d$. We also prove a
Berezin-Li-Yau inequality for domains contained in the hemisphere $\mathbb
S^2_+$. Moreover, we consider polyharmonic operators for which we prove
analogous results that highlight the role of dimension for P\'olya-type
inequalities. Finally, we provide sum rules for Laplacian eigenvalues on
spheres and compact two-point homogeneous spaces.
