  Tree-cut width is a graph parameter introduced by Wollan that is an analogue
of treewidth for the immersion order on graphs in the following sense: the
tree-cut width of a graph is functionally equivalent to the largest size of a
wall that can be found in it as an immersion. In this work we propose a variant
of the definition of tree-cut width that is functionally equivalent to the
original one, but for which we can state and prove a tight duality theorem
relating it to naturally defined dual objects: appropriately defined brambles
and tangles. Using this result we also propose a game characterization of
tree-cut width.
