  We revisit and extend the Durdevic theory of complete calculi on quantum
principal bundles. In this setting one naturally obtains a graded Hopf-Galois
extension of the higher order calculus and an intrinsic decomposition of degree
1-forms into horizontal and vertical forms. This proposal is appealing, since
it is consistently equipped with a canonical braiding and exactness of the
Atiyah sequence is guaranteed. Moreover, we provide examples of complete
calculi, including the noncommutative 2-torus, the quantum Hopf fibration and
differential calculi on crossed product algebras.
