  We present expressions for the matrix elements of the spin--spin operator
$\vec S_{\rm n}\cdot\vec S_{\rm p}$ in a variety of coupling schemes. These
results are then applied to calculate the expectation value $\langle\vec S_{\rm
n}\cdot\vec S_{\rm p}\rangle$ in eigenstates of a schematic Hamiltonian
describing neutrons and protons interacting in a single-$l$ shell through a
Surface Delta Interaction. The model allows us to trace $\langle\vec S_{\rm
n}\cdot\vec S_{\rm p}\rangle$ as a function of the competition between the
isovector and isoscalar interaction strengths and the spin--orbit splitting of
the $j=l\pm \frac{1}{2}$ shells. We find negative $\langle\vec S_{\rm
n}\cdot\vec S_{\rm p}\rangle$ values in the ground state of all even--even
$N=Z$ nuclei, contrary to what has been observed in hadronic inelastic
scattering at medium energies. We discuss the possible origin of this
discrepancy and indicate directions for future theoretical and experimental
studies related to neutron--proton spin--spin correlations.
