Covariant first order differential calculus on quantum Euclidean spheres

We study covariant differential calculus on the quantum spheres S_q^{N-1} which are quantum homogeneous spaces with coactions of the quantum groups O_q(N). The first part of the paper is devoted to first order differential calculus. A classification result is proved which says that for N>=6 there exist exactly two covariant first order differential calculi on S_q^{N-1} which satisfy the classification constraint that the bimodule of one-forms is generated as a free left module by the differentials of the generators of S_q^{N-1}. Both calculi exist also for 3<=N<=5. The same calculi can be constructed using a method introduced by Hermisson. In case N=3, the result is in accordance with the known result by Apel and Schm\"udgen for the Podles sphere. In the second part, higher order differential calculus and symmetry are treated. The relations which hold for the two-forms in the universal higher order calculus extending one of the two first order calculi are given. A "braiding" homomorphism is found. The existence of an upper bound for the order of differential forms is discussed.