Genomic rearrangements, encompassing mutational changes in the genome such as insertions, deletions, or inversions, are essential for genetic diversity. These rearrangements are typically orchestrated by enzymes involved in fundamental DNA repair processes such as homologous recombination or in the transposition of foreign genetic material by viruses and mobile genetic elements (MGEs). We report that IS110 insertion sequences, a family of minimal and autonomous MGEs, express a structured non-coding RNA that binds specifically to their encoded recombinase. This bridge RNA contains two internal loops encoding nucleotide stretches that base-pair with the target DNA and donor DNA, which is the IS110 element itself. We demonstrate that the target-binding and donor-binding loops can be independently reprogrammed to direct sequence-specific recombination between two DNA molecules. This modularity enables DNA insertion into genomic target sites as well as programmable DNA excision and inversion. The IS110 bridge system expands the diversity of nucleic acid-guided systems beyond CRISPR and RNA interference, offering a unified mechanism for the three fundamental DNA rearrangements required for genome design.