  Let $(G,K)$ be a Gelfand pair, with $G$ a Lie group of polynomial growth, and
let $\Sigma\subset{\mathbb R}^\ell$ be a homeomorphic image of the Gelfand
spectrum, obtained by choosing a generating system $D_1,\dots,D_\ell$ of
$G$-invariant differential operators on $G/K$ and associating to a bounded
spherical function $\varphi$ the $\ell$-tuple of its eigenvalues under the
action of the $D_j$'s.
  We say that property (S) holds for $(G,K)$ if the spherical transform maps
the bi-$K$-invariant Schwartz space ${\mathcal S}(K\backslash G/K)$
isomorphically onto ${\mathcal S}(\Sigma)$, the space of restrictions to
$\Sigma$ of the Schwartz functions on ${\mathbb R}^\ell$. This property is
known to hold for many nilpotent pairs, i.e., Gelfand pairs where $G=K\ltimes
N$, with $N$ nilpotent.
  In this paper we enlarge the scope of this analysis outside the range of
nilpotent pairs, stating the basic setting for general pairs of polynomial
growth and then focussing on strong Gelfand pairs.
