The symmetric calculus LG without interaction principles is context-free, as shown in Bastenhof 2010.  For the system with the interaction principles of §3.3, Melissen (2009) shows that all languages which are the intersection of a context- free language and the permutation closure of a context-free language are recognizable in LG. In this class, we find generalized forms of MIX (the language consisting of an equal number of symbols a, b, c, in any order), with equal multiplicity of k alphabet symbols in any order, and counting dependencies  an1…ank for any number k of alphabet symbols.  Patterns of this type are recognized by Range Concatenation Grammars and Global Index Grammars; a comparison with these formalisms then might be useful to fix the upper bound of the recognizing capacity of LG, which is as yet unknown.