  Symmetric fix-free codes are prefix condition codes in which each codeword is
required to be a palindrome. Their study is motivated by the topic of joint
source-channel coding. Although they have been considered by a few communities
they are not well understood. In earlier work we used a collection of instances
of Boolean satisfiability problems as a tool in the generation of all optimal
binary symmetric fix-free codes with n codewords and observed that the number
of different optimal codelength sequences grows slowly compared with the
corresponding number for prefix condition codes. We demonstrate that all
optimal symmetric fix-free codes can alternatively be obtained by sequences of
codes generated by simple manipulations starting from one particular code. We
also discuss simplifications in the process of searching for this set of codes.
