  It has been suggested that a colour-entanglement effect exists in the
Drell-Yan cross section for the 'double T-odd' contributions at low transverse
momentum $Q_T$, rendering the colour structure different from that predicted by
the usual factorisation formula [1]. These T-odd contributions can come from
the Boer-Mulders or Sivers transverse momentum dependent distribution
functions. The different colour structure should be visible already at the
lowest possible order that gives a contribution to the double Boer-Mulders
(dBM) or double Sivers (dS) effect, that is at the level of two gluon
exchanges. To discriminate between the different predictions, we compute the
leading-power contribution to the low-$Q_T$ dBM cross section at the two-gluon
exchange order in the context of a spectator model. The computation is
performed using a method of regions analysis with Collins subtraction terms
implemented. The results conform with the predictions of the factorisation
formula. In the cancellation of the colour entanglement, diagrams containing
the three-gluon vertex are essential. Furthermore, the Glauber region turns out
to play an important role - in fact, it is possible to assign the full
contribution to the dBM cross section at the given order to the region in which
the two gluons have Glauber scaling. A similar disentanglement of colour is
found for the dS effect.
