In this talk I will analyze composition methods with complex coefficients exhibiting the so-called “symmetry-conjugate” pattern in their distribution. In particular, I will study their behavior with respect to preservation of qualitative properties when projected on the real axis and how they compare with the usual left-right palindromic compositions. New schemes within this family up to order 8 will be proposed and illustrated on several examples. Some of the special features of this class of methods will also be reviewed.