Three types of 4-bar linkages can trace the same coupler curve according to the cognate mechanism law. Based on the investigation of the relationship between dimensions of a linkage and the harmonic component characteristic parameters of the coupler curve of the linkage, it was found that six types of linkages are related to the same coupler curve, and they belong to two groups of cognate mechanisms, in which the coupler curves have the same shape and are symmetric. The relationships among the dimensional types of these six mechanisms are presented. The corresponding examples have also proved these conclusions. This work enriches the cognate mechanism law, and can result in more candidate mechanisms for mechanism dimensional synthesis.