The present paper is devoted to the question what conclusions about invariant building blocks (maximal conserved moieties) in chemical reaction systems can be derived from the stoichiometry of these systems. We define a concept of maximal conserved-moiety vectors in terms of those vectors that possess four essential properties: conservation and integer-element properties, non-negativity, and non-decomposability, and examine the correspondence of these vectors to physically meaningful molecular units. In particular, we show that the set of all maximal conserved-moiety vectors (which turns out to be the complete set of fundamental non-negative solutions to a system of linear diophantine equations) corresponds to the set of all potential maximal conserved moieties admissible for the given stoichiometry. Our interpretation of the term moiety to express predominantly conservation of the empirical formula (rather than invariability of the bond structure) is explained and discussed. We propose and substantiate an algorithm for computing all maximal conserved-moiety vectors, and compare it with existing procedures for analysing the moiety structure. The number of these vectors may be equal to, less than, or greater than, the number of linearly independent conservation relations. A number of chemical and biochemical examples illustrate our analysis.