This paper deals with a refinement of the Bruhat cell decomposition for the generalized flag manifold \(G/B\) of a semisimple algebraic group \(G\). It is obtained by intersecting a given Bruhat cell with all cells of the dual decomposition provided by the orbits of the opposite Borel subgroup \(B_-\) on \(G/B\). The intersections decompose themselves as unions of products of affine spaces with tori whose respective dimensions are combinatorially expressed in terms of the Weyl group \(W\) of \(G\). The results can be used to derive explicit formulas for certain polynomials related to the Kazhdan-Lusztig polynomials and to prove the \(L\)-shellability of the Bruhat order on \(W\).