This paper attempts to resolve a debate on the relations between genealogy and kin classification. One theory from Lounsbury and Scheffler proposes that rewrite ruler (or L-rules) act on focal kintypes to define classes for labelling by the kin terms. A second approach from Lehman begins with a dimensioned genealogical space which becomes deformed into the kin term structure. The claim here is that for the Australian context a modified version of the second approach is the correct interpretation. Kinship systems are examples of a cultural strategy more consonant with Lehman than with Scheffler. The conceptual structures of kinship are multiple mathematical systems. They are built upon a small number of generating actions in a universal space of genealogical reckonings. For Australia, the extra reductions which govern the equivalence rules are symmetrical. The case is argued by tracing various implications of the Australian symmetries. They specify which equivalences can be derived from each other via isomorphisms and homomorphisms of the relevant group-theoretic models. A comparison of predicted class compositions and alignments against the Kimberley and other kinship systems provides qualified support for the claim.