The starting point of this article is a decades-old yet little-noticed sufficient condition, presented by Sassenfeld in 1951, for the convergence of the classical Gauss-Seidel method. The purpose of the present paper is to shed new light on Sassenfeld's criterion and to demonstrate that the original work can be perceived as a special case of a far more extensive concept in the context of preconditioners and iterative linear solvers. Our main result is a classification theorem for the set of all matrices which this general framework applies to.