  An experiment to measure single-spin asymmetries in semi-inclusive production
of charged pions in deep-inelastic scattering on a transversely polarized
$^3$He target was performed at Jefferson Lab in the kinematic region of
$0.16<x<0.35$ and $1.4<Q^2<2.7$ ${\rm GeV^2}$. The pretzelosity asymmetries on
$^3$He, which can be expressed as the convolution of the $h^\perp_{1T}$
transverse momentum dependent distribution functions and the Collins
fragmentation functions in the leading order, were measured for the first time.
Using the effective polarization approximation, we extracted the corresponding
neutron asymmetries from the measured $^3$He asymmetries and cross-section
ratios between the proton and $^3$He. Our results show that for both
$\pi^{\pm}$ on $^3$He and on the neutron the pretzelosity asymmetries are
consistent with zero within experimental uncertainties.
