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

Liberations and twists of real and complex spheres