  We discuss the prominent role played by bilateral symmetry and modified
Pascal triangles in self twin games, a subset of constant sum homogeneous
weighted majority games. We show that bilateral symmetry of the free
representations unequivocally identifies and characterizes this class of games
and that modified Pascal triangles describe their cardinality for combinations
of m and k, respectively linked through linear transforms to the key parameters
n, number of players and h, number of types in the game. Besides, we derive the
whole set of self twin games in the form of a genealogical tree obtained
through a simple constructive procedure in which each game of a generation,
corresponding to a given value of m, is able to give birth to one child or two
children (depending on the parity of m), self twin games of the next
generation. The breeding rules are, given the parity of m, invariant through
generations and quite simple.
