  We experimentally and theoretically investigate the influence of the
dipole-dipole interactions (DDIs) on the total inter-species interaction in an
erbium-dysprosium mixture. By rotating the dipole orientation we are able to
tune the effect of the long-range and anisotropic DDI, and therefore the
in-trap clouds displacement. We present a theoretical description for our
binary system based on an extended Gross-Pitaevskii (eGP) theory, including the
single-species beyond mean-field terms, and we predict a lower and an upper
bound for the inter-species scattering length $a_{12}$. Our work is a first
step towards the investigation of the experimentally unexplored dipolar
miscibility-immiscibility phase diagram and the realization of quantum droplets
and supersolid states with heteronuclear dipolar mixtures.
