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 $H_1^{perp}$ 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 $H_1^{perp}$. 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.
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