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 $Deltasigma_L$ and $Deltasigma_T$ 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.
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