  We study the 10 noncommutative spheres obtained by liberating, twisting, and
liberating+twisting the real and complex spheres $S^{N-1}_\mathbb
R,S^{N-1}_\mathbb C$. At the axiomatic level, we show that, under very strong
axioms, these 10 spheres are the only ones. Our main results concern the
computation of the quantum isometry groups of these 10 spheres, taken in an
affine real/complex sense. We formulate as well a proposal for an extended
formalism, comprising 18 spheres.
