  The spherical centroid body of a centrally-symmetric convex body in the
Euclidean unit sphere is introduced. Two alternative definitions - one
geometric, the other probabilistic in nature - are given and shown to lead to
the same objects. The geometric approach is then used to establish a number of
basic properties of spherical centroid bodies, while the probabilistic approach
inspires the proof of a spherical analogue of the classical polar
Busemann-Petty centroid inequality.
