This work reports the first observation of azimuthal asymmetries around the thrust axis in e(+)e(-) annihilation of pairs of back-to-back charged pions in one hemisphere, and pi(0) and eta mesons in the opposite hemisphere. These results are complemented by a new analysis of pairs of back-to-back charged pions. The pi(0) and eta asymmetries rise with the relative momentum z of the detected hadrons as well as with the transverse momentum with respect to the thrust axis. These asymmetries are sensitive to the Collins fragmentation function H-1(perpendicular to) and provide complementary information to previous measurements with charged pions and kaons in the final state. In particular, the eta final states will provide additional information on the flavor structure of H-1(perpendicular to). This is the first measurement of the explicit transverse-momentum dependence of the Collins fragmentation function from Belle data. It uses a dataset of 980.4 fb(-1) collected by the Belle experiment at or near a center-of-mass energy of 10.58 GeV.