  We show that all the tangles in a finite graph or matroid can be
distinguished by a single tree-decomposition that is invariant under the
automorphisms of the graph or matroid. This comes as a corollary of a similar
decomposition theorem for more general combinatorial structures, which has
further applications.
  These include a new approach to cluster analysis and image segmentation. As
another illustration for the abstract theorem, we show that applying it to
edge-tangles yields the Gomory-Hu theorem.
