We investigate the double spin asymmetries of pion production in semi-inclusive deep inelastic scattering with a longitudinal polarized beam off a transversely polarized proton target. Particularly, we consider the cos φS and cos(2φ h − φS) modulations, which can be interpreted by the convolution of the twist-3 transverse momentum dependent distributions and twist-2 fragmentation functions. Three different origins are taken into account simultaneously for each asymmetry: the gT D1 term, the eT H , and e ⊥ T (x, k 2 T ) in a spectator-diquark model including vector diquarks. Then we predict the two corresponding asymmetries for charged and neutral pions at the kinematics of HERMES, JLab, and COMPASS for the first time. The numerical estimates indicate that the two different angular-dependence asymmetries are sizable by several percent at HERMES and JLab, and the cos φS asymmetry has a strong dependence on the Bjorken x. Our predictions also show that the dominant contribution to the cos φS asymmetry comes from the gT D1 term, while the g ⊥ T D1 term gives the main contribution to the cos(2φ h − φS) asymmetry; the other two T -odd terms almost give negligible contributions. In particular, the cos(2φ h − φS) asymmetry provides a unique opportunity to probe the distribution g ⊥ T .