  We analyze composition methods with complex coefficients exhibiting the
so-called ``symmetry-conjugate'' pattern in their distribution. In particular,
we study their behavior with respect to preservation of qualitative properties
when projected on the real axis and we compare them with the usual left-right
palindromic compositions. New schemes within this family up to order 8 are
proposed and their efficiency is tested on several examples. Our analysis shows
that higher-order schemes are more efficient even when time step sizes are
relatively large.
