  We prove explicit upper and lower bounds for the Poisson hierarchy, the
averaged $L^1$-moment spectra
$\{\dfrac{\mathcal{A}_k\left(B_R^M\right)}{\text{vol}\left(S_R^M\right)}\}_{k=1}^\infty$,
and the torsional rigidity $\mathcal{A}_1(B^M_R)$ of a geodesic ball $B^M_R$ in
a Riemannian manifold $M^n$ which satisfies that the mean curvatures of the
geodesic spheres $S^M_r$ included in it, (up to the boundary $S^M_R$), are
controlled by the radial mean curvature of the geodesic spheres
$S^\omega_r(o_\omega)$ with same radius centered at the center $o_\omega$ of a
rotationally symmetric model space $M^n_\omega$. As a consecuence, we prove a
first Dirichlet eigenvalue $\lambda_1(B^M_R)$ comparison theorem and show that
equality with the bound $\lambda_1(B^\omega_R(o_\omega))$, (where
$B^\omega_r(o_\omega)$ is the geodesic $r$-ball in $M^n_\omega$), characterizes
the $L^1$-moment spectrum $\{\mathcal{A}_k(B^M_R)\}_{k=1}^\infty$ as the
sequence $\{\mathcal{A}_k(B^\omega_R)\}_{k=1}^\infty$ and vice-versa.
