  We report the first measurement of the double-spin asymmetry $A_{LT}$ for
charged pion electroproduction in semi\nobreakdash-inclusive
deep\nobreakdash-inelastic electron scattering on a transversely polarized
$^{3}$He target. The kinematics focused on the valence quark region,
$0.16<x<0.35$ with $1.4<Q^{2}<2.7\,\textrm{GeV}^{2}$. 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 $\pi^{-}$ production on $^{3}$He and the neutron, while our $\pi^{+}$
asymmetries are consistent with zero.
