  We study aspects of the Wasserstein distance in the context of self-similar
measures. Computing this distance between two measures involves minimising
certain moment integrals over the space of \emph{couplings}, which are measures
on the product space with the original measures as prescribed marginals. We
focus our attention on self-similar measures associated to equicontractive
iterated function systems satisfying the open set condition and consisting of
two maps on the unit interval. We are particularly interested in understanding
the restricted family of \emph{self-similar} couplings and our main achievement
is the explicit computation of the 1st and 2nd moment integrals for such
couplings. We show that this family is enough to yield an explicit formula for
the 1st Wasserstein distance and provide non-trivial upper and lower bounds for
the 2nd Wasserstein distance.
