  Let $G/H$ be a reductive symmetric space over a $p$-adic field $F$, the
algebraic groups $G$ and $H$ being assumed semisimple of relative rank $1$. One
of the branching problems for the Steinberg representation $\St_G$ of $G$ is
the determination of the dimension of the intertwining space ${\rm Hom}_H
(\St_G ,\pi )$, for any irreducible representation $\pi$ of $H$. In this work
we do not compute this dimension, but show how it is related to the dimensions
of some other intertwining spaces ${\rm Hom}_{K_i} ({\tilde \pi} ,1)$, for a
certain finite family $K_i$, $i=1,...,r$, of anisotropic subgroups of $H$ (here
${\tilde \pi}$ denote the contragredient representation, and $1$ the trivial
character). In other words we show that there is a sort of `reciprocity law'
relating two different branching problems.
