  We show that for a metric space with an even number of points there is a
1-Lipschitz map to a tree-like space with the same matching number. This result
gives the first basic version of an unoriented Kantorovich duality. The study
of the duality gives a version of global calibrations for 1-chains with
coefficients in $\mathbb Z_2$. Finally we extend the results to infinite metric
spaces and present a notion of "matching dimension" which arises naturally.
