We report the first measurement of the target single-spin asymmetry, Ay, in quasielastic scattering from the inclusive reaction He3↑(e,e′) on a He3 gas target polarized normal to the lepton scattering plane. Assuming time-reversal invariance, this asymmetry is strictly zero for one-photon exchange. A nonzero Ay 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 Q2=0.13, 0.46, and 0.97GeV2. These measurements demonstrate, for the first time, that the He3 asymmetry is clearly nonzero and negative at the 4σ-9σ level. Using measured proton-to-He3 cross-section ratios and the effective polarization approximation, neutron asymmetries of -(1-3)% were obtained. The neutron asymmetry at high Q2 is related to moments of the generalized parton distributions (GPDs). Our measured neutron asymmetry at Q2=0.97GeV2 agrees well with a prediction based on two-photon exchange using a GPD model and thus provides a new, independent constraint on these distributions.