We use Stein characterizations to obtain new moment-type estimators for the parameters of three classical spherical distributions (namely the Fisher-Bingham, the von Mises-Fisher, and the Watson distributions) in the i.i.d. case. This leads to explicit estimators which have good asymptotic properties (close to efficiency) and therefore lead to interesting alternatives to classical maximum likelihood methods or more recent score matching estimators. We perform competitive simulation studies to assess the quality of the new estimators. Finally, the practical relevance of our estimators is illustrated on a real data application in spherical latent representations of handwritten numbers.