  At present, the Boer-Mulders (BM) function for a given quark flavour is
extracted from data on semi-inclusive deep inelastic scattering (SIDIS) using
the simplifying assumption that it is proportional to the Sivers function for
that flavour. In a recent paper we suggested that the consistency of this
assumption could be tested using information on so-called difference
asymmetries, i.e. the difference between the asymmetries in the production of
particles and their anti-particles.
  In this paper, using the SIDIS COMPASS deuteron data on the
$\langle\cos\phi_h\rangle$, $\langle\cos 2 \phi_h\rangle$ and Sivers difference
asymmetries, we carry out two independent consistency tests of the assumption
of proportionality, but here applied to the sum of the valence-quark
contributions. We find that such an assumption is compatible with the data. We
also show that the proportionality assumptions made in the existing
parametrizations of the BM functions are not compatible with our analysis,
which suggests that the published results for the Boer-Mulders functions for
individual flavours are unreliable.
  The $\langle\cos\phi_h\rangle$ and $\langle\cos 2 \phi_h\rangle$ asymmetries
receive contributions also from the, in principle, calculable Cahn effect. We
succeed in extracting the Cahn contributions from experiment (we believe for
the first time) and compare with their calculated values, with interesting
implications.
