In a previous paper [ibid. 122, No. 3, 569-609 (2009; Zbl 1185.37036)] the authors introduced the concept of block words and resonant chains. This paper constitutes a continuation of the previous one, addressing the following four issues: the introduction of the concept of parity to characterize periodic orbits; a detailed study of the symbol plane; a clarification of the behavior of symmetric periodic orbits in the resonant chain and a determination of their block word sequence; and an improved construction method to make a braid for a non-Birkhoff periodic orbit in a resonance chain.