  A set of measurements of azimuthal asymmetries in the production of pairs of
identified hadrons in deep-inelastic scattering of muons on transversely
polarised $^6$LiD (deuteron) and NH$_3$ (proton) targets is presented. All
available data collected in the years 2002-2004 and 2007/2010 with the COMPASS
spectrometer using a muon beam of 160 GeV/$c$ at the CERN SPS were analysed.
The asymmetries provide access to the transversity distribution functions via a
fragmentation function that in principle may be independently obtained from
$e^+e^-$ annihilation data. Results are presented, discussed and compared to
existing measurements as well as to model predictions. Asymmetries of
$\pi^{+}\pi^{-}$ pairs measured with the proton target as a function of the
Bjorken scaling variable are sizeable in the range $x> 0.032$, indicating
non-vanishing transversity distribution and di-hadron interference
fragmentation functions. As already pointed out by several authors, the small
asymmetries of $\pi^{+}\pi^{-}$ measured on the $^6$LiD target can be
interpreted as indication for a cancellation of $u$ and $d$-quark transversity
distributions.
