Asymmetry within both the amplification factor values (Ak) and cross-catalytic hypercyclic constant (Kjk) and its influence on the stability of a two-membered error-prone hypercycle has been exhaustively studied from a deterministic point of view and the bifurcation diagram as a function of the quality factor (Q) has been obtained. In the more general case, several Q critical values appear, changing their relative position in the diagram depending on the Ak and Kjk values. The order of the Q critical values affects both the general properties of the system and the stability of the hypercyclic organization. The importance of this asymmetry in the selective and evolutionary properties of the hypercycle is also discussed.