  We describe noncommutative geometric aspects of twisted deformations, in
particular of the spheres in Connes and Landi [8] and in Connes and Dubois
Violette [7], by using the differential and integral calculus on these spaces
that is covariant under the action of their corresponding quantum symmetry
groups. We start from multiparametric deformations of the orthogonal groups and
related planes and spheres. We show that only in the twisted limit of these
multiparametric deformations the covariant calculus on the plane gives by a
quotient procedure a meaningful calculus on the sphere. In this calculus the
external algebra has the same dimension of the classical one. We develop the
Haar functional on spheres and use it to define an integral on forms. In the
twisted limit (differently from the general multiparametric case) the Haar
functional is a trace and we thus obtain a cycle on the algebra. Moreover we
explicitely construct the *-Hodge operator on the space of forms on the plane
and then by quotient on the sphere. We apply our results to even spheres and we
compute the Chern-Connes pairing between the character of this cycle, i.e. a
cyclic 2n-cocycle, and the instanton projector defined in [7].
