In this article we examine the saturation conjecture on decompositions of tensor products of irreducible representations for complex semisimple algebraic groups of type D (the even spin groups: Spin for an integer), extending work done by Kumar–Kapovich–Millson on Spin(8). Our main theorem asserts that the saturation conjecture holds for Spin(10) and Spin(12): for all triples of dominants weights such that is in the root lattice, and for any N > 0, if and only iffor or . Some related results for groups of other types are listed as well.