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We investigate the double spin asymmetries of pion production insemi-inclusive deep inelastic scattering with a longitudinal polarized beam offa transversely polarized proton target. Particularly, we consider the$\cos\phi_S$ and $\cos(2\phi_h -\phi_S)$ modulations, which can be interpretedby the convolution of the twist-3 transverse momentum dependent distributionsand twist-2 fragmentation functions. Three different origins are taken intoaccount simultaneously for each asymmetry: the $g_T D_1$ term, the $e_TH_1^\perp$ term, and the $e_T^\perp H_1^\perp$ term in the $\cos\phi_S$asymmetry; and the $g_T^\perp D_1$ term, the $e_T H_1^\perp$ term, and the$e_T^\perp H_1^\perp$ term in the $\cos(2\phi_h -\phi_S)$ asymmetry. Wecalculate the four twist-3 distributions $g_T(x,\boldsymbol{k}_T^2)$,$g_T^\perp(x,\boldsymbol{k}_T^2)$, $e_T(x,\boldsymbol{k}_T^2)$, and$e_T^\perp(x,\boldsymbol{k}_T^2)$ in a spectator-diquark model including vectordiquarks. Then we predict the two corresponding asymmetries for charged andneutral pions at the kinematics of HERMES, JLab, and COMPASS for the firsttime. The numerical estimates indicate that the two differentangular-dependence asymmetries are sizable by several percent at HERMES andJLab, and the $\cos\phi_S$ asymmetry has a strong dependence on the Bjorken$x$. Our predictions also show that the dominant contribution to the$\cos\phi_S$ asymmetry comes from the $g_T D_1$ term, while the $g_T^\perp D_1$term gives the main contribution to the $\cos(2\phi_h -\phi_S)$ asymmetry; theother two $T$-odd terms almost give negligible contributions. In particular,the $\cos(2\phi_h -\phi_S)$ asymmetry provides a unique opportunity to probethe distribution $g_T^\perp$.

Double Spin Asymmetries $A_{LT}^{\cos\phi_S}$ and $A_{LT}^{\cos(2\phi_h -\phi_S)}$ in semi-inclusive DIS