The crystal structure of cumengéite, Pb₂₁Cu₂₀Cl₄₂(OH)₄₀, tetragonal, a= 15.065(2), c= 24.436(5) Å, V = 5546(2) ų, space group I4/mmm, Z = 2, has been solved by direct methods and refined by least-squares to an R index of 7.1 % for 1158 observed (I > 2.5σ I) reflections. The cumengéite structure shows a very diverse range of cation coordinations. There are five distinct Pb sites with the following coordination polyhedra: octahedron, square antiprism, augmented trigonal prism, a very distorted biaugmented trigonal prism and a fairly regular biaugmented trigonal prism, the latter being only half-occupied; there are two distinct Cu sites with octahedral and square pyramidal coordination respectively. The coordination polyhedra share elements to form prominent columns or rods of polyhedra parallel to the c-axis and centred on the 4-fold axes of the unit cell. These rods form a body-centred array, and link by sharing elements of their coordination polyhedra.