We consider skew metrics (equivalently, transitive relations that are tournaments, linear orderings) valued in Sugihara semigroups on autodual chains. We prove that, for odd chains and chains without a unit, skew metrics classify certain tree-like structures that we call perfect augmented plane towers. When the chain is finite and has cardinality 2K + 1, skew metrics on a set X give rise to perfect rooted plane trees of height K whose frontier is a linear preorder of X. Any linear ordering on X gives rise to an ordering on the set of its skew metrics valued in an arbitrary involutive residuated lattice Q. If Q satisfies the mix rule, then this poset is most often a lattice. We study this lattice for X = {1,. .. , n} and Q the Sugihara monoid on the chain of cardinality 2K + 1. We give a combinatorial model of this lattice by describing its covers as moves on a space of words coding perfect augmented plane trees. Using the combinatorial model, we develop enumerative considerations on this lattice.