  We present a geometric description, based on the affine Weyl group
E_{6}^{(1)}, of two discrete analogues of the Painlev\'e VI equation, known as
the asymmetric q-P_{V} and asymmetric d-P_{IV}. This approach allows us to
describe in a unified way the evolution of the mapping along the independent
variable and along the various parameters (the latter evolution being the one
induced by the Schlesinger transformations). It turns out that both discrete
Painlev\'e equations exhibit the property of self-duality: the same equation
governs the evolution along any direction in the space of E_{6}^{(1)}.
