  The cross section for back-to-back hadron pair production in $e^+e^-$
annihilation provides access to the dihadron fragmentation functions (DiFF)
needed to extract nucleon parton distribution functions from the semi-inclusive
deep inelastic scattering (SIDIS) experiments with two detected final state
hadrons. Particular attention is given to the so-called interference DiFF
(IFF), which makes it possible to extract the transversity parton distribution
of the nucleon in the collinear framework. However, previously unnoticed
discrepancies were recently highlighted between the definitions of the IFFs
appearing in the collinear kinematics when reconstructed from DiFFs entering
the unintegrated fully differential cross sections of SIDIS and $e^+e^-$
annihilation processes. In this work, to clarify this problem we rederive the
fully differential cross section for $e^+e^-$ annihilation at the leading-twist
approximation. We find a mistake in the definition of the kinematics in the
original expression that systematically affects a subset of terms and that
leads to two significant consequences. First, the discrepancy between the IFF
definitions in the cross sections for SIDIS and $e^+e^-$ annihilation is
resolved. Second, the previously derived azimuthal asymmetry for accessing the
helicity dependent DiFF $G_1^\perp$ in $e^+e^-$ annihilation vanishes, which
explains the nonobservation of this asymmetry in the recent experimental
searches by the ${\tt BELLE}$ Collaboration. We discuss the recently proposed
alternative option to extract $G_1^\perp$.
