Partitions of complete twisted graphs into plane spanning trees

We characterize all partitions of the complete twisted graph $T_{2n}$ into plane spanning trees. In the case of partitions of $T_{2n}$ into isomorphic plane spanning trees, we show that all trees in these partitions must be balanced double stars. As a consequence of our results, any complete topological graph with $n$ vertices contains a complete topological subgraph with $m\ge c{\log }^{1\mathrm{/}8}n$ vertices that admits a partition into plane spanning trees.