We report the first measurement of the double-spin asymmetry A{LT} for charged pion electroproduction in semi-inclusive deep-inelastic electron scattering on a transversely polarized {3}He target. The kinematics focused on the valence quark region, 0.16<x<0.35 with 1.4<Q{2}<2.7 GeV{2}. The corresponding neutron A{LT} asymmetries were extracted from the measured {3}He asymmetries and proton over {3}He cross section ratios using the effective polarization approximation. These new data probe the transverse momentum dependent parton distribution function g{1T}{q} and therefore provide access to quark spin-orbit correlations. Our results indicate a positive azimuthal asymmetry for π{-} production on {3}He and the neutron, while our π{+} asymmetries are consistent with zero.