This paper is about two arrangements of hyperplanes. The first --- the Shi
arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig
cells in the affine Weyl group of type $A$. The second --- the Ish arrangement
--- was recently defined by the first author who used the two arrangements
together to give a new interpretation of the $q,t$-Catalan numbers of Garsia
and Haiman. In the present paper we will define a mysterious "combinatorial
symmetry" between the two arrangements and show that this symmetry preserves a
great deal of information. For example, the Shi and Ish arrangements share the
same 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 complete
graph. Our proofs are based on nice combinatorial labelings of Shi and Ish
regions and a new set-valued statistic on these regions.