  We study sequences of noncommutative random variables which are invariant
under "quantum transformations" coming from an orthogonal quantum group
satisfying the "easiness" condition axiomatized in our previous paper. For 10
easy quantum groups, we obtain de Finetti type theorems characterizing the
joint distribution of any infinite quantum invariant sequence. In particular,
we give a new and unified proof of the classical results of de Finetti and
Freedman for the easy groups S_n, O_n, which is based on the combinatorial
theory of cumulants. We also recover the free de Finetti theorem of K\"ostler
and Speicher, and the characterization of operator-valued free semicircular
families due to Curran. We consider also finite sequences, and prove an
approximation result in the spirit of Diaconis and Freedman.
