Let \(G\) be a real reductive group, and fix an infinitesimal character \(\lambda\) for \(G\). The set of equivalence classes of irreducible Harish-Chandra modules for \(G\) with infinitesimal character \(\lambda\) is finite, and hence the vector space \({\mathcal G}(\lambda)\) spanned by the characters of these representations is finite-dimensional. Suppose \(\lambda\) is regular and integral. Then \({\mathcal G}(\lambda)\) carries a representation of the Weyl group \(W\) of \(G\). This representation plays a crucial role in the program of computing all the irreducible characters of \(G\), and it is important to understand this module structure. This paper is concerned with the basic part of this problem, which is the structure of the “cells” of \({\mathcal G} (\lambda)\), the minimal subquotients of \({\mathcal G}(\lambda)\) spanned by irreducible characters. Such a cell is identified with a set of irreducible Harish-Chandra modules. The main tool is to relate cells to similarly defined cells in category \({\mathcal O}\), which are better understood. The main result is a computation of the module structure of “most” cells for classical groups.