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 \(m \ge c\log ^{1/8} n\) vertices that admits a partition into plane spanning trees.