We describe and characterize the contractively decomposable projections on noncommutative Lp-spaces. Our result relies on a new lifting result for decomposable maps of independent interest and on some tools from ergodic theory. Our theorem is new even for finite-dimensional Schatten spaces. Our description allows us to connect this topic with W⁎-ternary rings of operators and a slight generalization of our result for more general projections makes JBW⁎-triples appear in this context. We also prove that all rectangular Lp-spaces associated with W⁎-ternary rings of operators arise as contractively decomposable complemented subspaces of noncommutative Lp-spaces. Finally, we introduce a notion of Lp-space associated to each σ-finite JBW⁎-triple and we explain the link with the context of this paper.