We report the first measurement of the target single-spin asymmetry, A(y), in quasielastic scattering from the inclusive reaction He-3(up arrow)(e,e') on a He-3 gas target polarized normal to the lepton scattering plane. Assuming time-reversal invariance, this asymmetry is strictly zero for one-photon exchange. A nonzero A(y) can arise from the interference between the one-and two-photon exchange processes which is sensitive to the details of the substructure of the nucleon. An experiment recently completed at Jefferson Lab yielded asymmetries with high statistical precision at Q(2) = 0.13, 0.46, and 0.97 GeV2. These measurements demonstrate, for the first time, that the He-3 asymmetry is clearly nonzero and negative at the 4 sigma-9 sigma level. Using measured proton-to-He-3 cross-section ratios and the effective polarization approximation, neutron asymmetries of -(1-3)% were obtained. The neutron asymmetry at high Q(2) is related to moments of the generalized parton distributions (GPDs). Our measured neutron asymmetry at Q(2) = 0.97 GeV2 agrees well with a prediction based on two-photon exchange using a GPD model and thus provides a new, independent constraint on these distributions.