Kullback and Leibler's minimum discrimination information function ∑ i=1 n p i log 2( p i q i ) , commonly known as the directed divergence in information theory, between the two probability distributions ( p 1, p 2,3., p n ) and ( q 1, q 2, 3., q n ), p i ≥ 0, q i ≥ 0, i = 1,2,3., n; Σ i n = 1 p i = Σ i n = 1 q i = 1, such that p i = 0 for all those indices i for which q i = 0, has its applications in various disciplines such as economics, international trade, pattern recognition, constructibility theory, decision theory, etc. Due to its diversified applications, it is desirable to have more information about its intuitive algebraic and analytic properties. Total symmetry (or simply symmetry) is one of its most important intuitive algebraic properties. From mathematical poin of view, it is desirable to weaken it in the strict sense. Such a strict weakening can be achieved in several ways. In this paper, the authors have proposed a new concept called ‘quasicyclic symmetry’, a strictly weaker form of symmetry and have utilized it in proving some characterization theorems concerning the directed divergence. The first two of these theorems take into consideration various aspects involving the use of the fundamental equation of the directed divergence and some information-theoretic ideas. The third theorem makes use of a number-theoretic equation.