Maps of 2D DNA walk of 671 examined chromosomes show composition complexity change from symmetrical half-turn in bacteria to pseudo-random trajectories in archaea, fungi and humans. In silico transformation of gene order and strand position returns most of the analyzed chromosomes to a symmetrical bacterial-like state with one transition point. The transformed chromosomal sequences also reveal remarkable segmental compositional symmetry between regions from different strands located equidistantly from the transition point. Despite extensive chromosome rearrangement the relation of gene numbers on opposite strands for chromosomes of different taxa varies in narrow limits around unity with Pearson coefficient r = 0.98. Similar relation is observed for total genes' length (r = 0.86) and cumulative GC (r = 0.95) and AT (r = 0.97) skews. This is also true for human coding sequences (CDS), which comprise only several percent of the entire chromosome length. We found that frequency distributions of the length of gene clusters, continuously located on the same strand, have close values for both strands. Eukaryotic gene distribution is believed to be non-random. Contribution of different subsystems to the noted symmetries and distributions, and evolutionary aspects of symmetry are discussed.