  Measurements of polarized neutron--polarized $^3$He scattering are reported.
The target consisted of cryogenically-polarized solid $^3$He, thickness 0.04
atom/b and polarization 40%. The longitudinal and transverse total
cross-section differences $\Delta\sigma_L$ and $\Delta\sigma_T$ were measured
for incident neutron energies 2-8 MeV. The results are compared to phase-shift
predictions based on four different analyses of n-$^3$He scattering. The best
agreement is obtained with a recent R-matrix analysis of A=4 scattering and
reaction data, lending strong suport to the $^4$He level scheme obtained in
that analysis.
