In this paper we revisit a classic HOS-based BSS criterion, namely the maximization of the higher-order moments of the estimated sources. The main contributions of this paper are: (i) a thorough study of the solutions given by popular HOS-based BSS criteria (including spurious solutions) and (ii) a method for estimating the source signals based on the eigendecomposition of certain adjustable HOS-matrices. Results are illustrated by computer simulations. r 2006 Elsevier B.V. All rights reserved.