We study heavy spin effects in $\bar{b}\bar{b}ud$ and $\bar{b}\bar{c}ud$ four-quark systems using the Born-Oppenheimer approximation and existing antistatic-antistatic potentials computed with lattice QCD. We report about a recent refined investigation of the $\bar{b}\bar{b}ud$ system with $I(J^P)=0(1^-)$, where we predicted a tetraquark resonance slightly above the $B^{*}B^{*}$ threshold. Furthermore, we extend our Born-Oppenheimer approach to $\bar{b}\bar{c}ud$ four-quark systems. For quantum numbers $I(J^P)=0(0^+)$ as well as $I(J^P)=0(1^+)$ we find virtual bound states rather far away from the lowest meson-meson thresholds.