  Permutation sorting, one of the fundamental steps in pre-processing data for
the efficient application of other algorithms, has a long history in
mathematical research literature and has numerous applications. Two
special-purpose sorting operations are considered in this paper: context
directed swap, abbreviated cds, and context directed reversal, abbreviated cdr.
These are special cases of sorting operations that were studied in prior work
on permutation sorting. Moreover, cds and cdr have been postulated to model
molecular sorting events that occur in the genome maintenance program of
certain species of single-celled organisms called ciliates.
  This paper investigates mathematical aspects of these two sorting operations.
The main result of this paper is a generalization of previously discovered
characterizations of cds-sortability of a permutation. The combinatorial
structure underlying this generalization suggests natural combinatorial
two-player games. These games are the main mathematical innovation of this
paper.
