Prokaryotic evolution takes place within and between genomes, when significant amounts of genes are transferred and recombined between interacting genetic partners. These non-tree-like evolutionary processes, intertwined with events of vertical descent, lead to a massive production of unrooted trees in which branches, nodes and groupings have different biological meanings than for the rooted trees usually studied by phylogenetics. Such unrooted gene trees can not only inform us about organismal phylogeny, but also about the variety of evolutionary, genetic, functional and ecological relationships affecting a plurality of evolutionary units, at multiple levels - from genes, groups of genes, organisms and consortia, to communities. Here we introduce new notions designed to analyze unrooted trees with more depth and accuracy. We demonstrate how a clanistic perspective can significantly improve our knowledge of evolutionary processes and relationships for most evolving systems, whether they are mobile genetic elements or cellular genomes.