  Of the many modern approaches to calculating evolutionary distance via models
of genome rearrangement, most are tied to a particular set of genomic modelling
assumptions and to a restricted class of allowed rearrangements. The "position
paradigm", in which genomes are represented as permutations signifying the
position (and orientation) of each region, enables a refined model-based
approach, where one can select biologically plausible rearrangements and assign
to them relative probabilities/costs. Here, one must further incorporate any
underlying structural symmetry of the genomes into the calculations and ensure
that this symmetry is reflected in the model. In our recently-introduced
framework of {\em genome algebras}, each genome corresponds to an element that
simultaneously incorporates all of its inherent physical symmetries. The
representation theory of these algebras then provides a natural model of
evolution via rearrangement as a Markov chain. Whilst the implementation of
this framework to calculate distances for genomes with `practical' numbers of
regions is currently computationally infeasible, we consider it to be a
significant theoretical advance: one can incorporate different genomic
modelling assumptions, calculate various genomic distances, and compare the
results under different rearrangement models. The aim of this paper is to
demonstrate some of these features.
