  Noncommutative analogues of n-dimensional balls are defined by repeated
application of the quantum double suspension to the classical low-dimensional
spaces. In the `even-dimensional' case they correspond to the Twisted Canonical
Commutation Relations of Pusz and Woronowicz. Then quantum spheres are
constructed as double manifolds of noncommutative balls. Both C*-algebras and
polynomial algebras of the objects in question are defined and analyzed, and
their relations with previously known examples are presented. Our construction
generalizes that of Hajac, Matthes and Szymanski for `dimension 2', and leads
to a new class of quantum spheres (already on the C*-algebra level) in all
`even-dimensions'.
