Inequalities between codon usage probabilities for quartets of codons are derived using a minimum principle for codon-anticodon interaction and a probability sum rule in the framework of the Crystal Basis Model of the genetic code. Performing this study separately for the Early and for the Eukaryotic Genetic Code, we observe a consistency in the obtained results for the two codes as well as a good agreement with data from Kasuza Codon Usage Database. Then we analyze the coherent changes of sign occurring, in the evolution from the Early to the Eukaryotic code, in the two parameters regulating the codon-anticodon interaction potential.