  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.
