Analysis of DNA composition at several length scales constitutes the bulk of many early studies aimed at unravelling the complexity of the organization and functionality of genomes. Dinucleotide relative abundances are considered an idiosyncratic feature of genomes, regarded as a 'genomic signature'. Motivated by this finding, we introduce the 'Generalized Genomic Signatures' (GGSs), composed of over- and under-abundances of all oligonucleotides of a given length, thus filtering out compositional trends and neighbour preferences at any shorter range. Previous works on alignment-free genomic comparisons mostly rely on k-mer frequencies and not on distance-dependent neighbour preferences. Therein, nucleotide composition and proximity preferences are combined, while in the present work they are strictly separated, focusing uniquely on neighbour relationships. GGSs retain the potential or even outperform genomic signatures defined at the dinucleotide level in distinguishing between taxonomic subdivisions of bacteria, and can be more effectively implemented in microbial phylogenetic reconstruction. Moreover, we compare DNA sequences from the human genome corresponding to protein coding segments, conserved non-coding elements and non-functional DNA stretches. These classes of sequences have distinctive GGSs according to their genomic role and degree of conservation. Overall, GGSs constitute a trait characteristic of the evolutionary origin and functionality of different genomic segments.