Coronaviruses (CoVs) have very large RNA viral genomes with a distinct genomic architecture of core and accessory open reading frames (ORFs). It is of utmost importance to understand their patterns and limits of homologous and non-homologous recombination, because such events may affect the emergence of novel CoV strains, alter their host range, infection rate, tissue tropism pathogenicity, and their ability to escape vaccination programs. Intratypic recombination among closely related CoVs of the same subgenus has often been reported; however, the patterns and limits of genomic exchange between more distantly related CoV lineages (intertypic recombination) needs further investigation. Here, we report computational/evolutionary analyses that clearly demonstrate a substantial ability for CoVs of different subgenera to recombine. Furthermore, we show that CoVs can obtain-through non-homologous recombination-accessory ORFs from core ORFs, exchange accessory ORFs with different CoV genera, with other viruses (i.e., toroviruses, influenza C/D, reoviruses, rotaviruses, astroviruses) and even with hosts. Intriguingly, most of these radical events result from double-crossovers surrounding the Spike ORF, thus highlighting both the instability and mobile nature of this genomic region. While many such events have often occurred during the evolution of various CoVs, the genomic architecture of the relatively young SARS-CoV/SARS-CoV-2 lineage so far appears to be stable.