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This paper is about two arrangements of hyperplanes. The first --- the Shiarrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztigcells in the affine Weyl group of type $A$. The second --- the Ish arrangement--- was recently defined by the first author who used the two arrangementstogether to give a new interpretation of the $q,t$-Catalan numbers of Garsiaand Haiman. In the present paper we will define a mysterious "combinatorialsymmetry" between the two arrangements and show that this symmetry preserves agreat deal of information. For example, the Shi and Ish arrangements share thesame characteristic polynomial, the same numbers of regions, bounded regions,dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom",etc. Moreover, all of these results hold in the greater generality of "deleted"Shi and Ish arrangements corresponding to an arbitrary subgraph of the completegraph. Our proofs are based on nice combinatorial labelings of Shi and Ishregions and a new set partition-valued statistic on these regions.

The Shi arrangement and the Ish arrangement