Newly replicated regions in genomes typically diverge in sequence over time, as random mutations accumulate and differential functions and selective pressures arise. Concerted sequence evolution in nuclear genomes occurs but is uncommon. Interestingly, mitochondrial genomes in several animal groups have duplicate or triplicate control regions or genes that undergo highly rapid concerted evolution and maintain nearly 100% sequence identity, including in seabirds, fishes, and cephalopods. These occurrences are poorly understood and are exceptional: 1) mitochondrial genomes generally lack replicate features despite many likely genome duplication events that give rise to gene rearrangements in diverse animals, 2) control regions generally evolve rapidly between individuals, making maintenance of sequence identity between paralogs that much more challenging, if not a contradiction, and 3) convergent sequence evolution in nuclear genomes is maintained by recombination but it is rare in mitochondrial genomes, suggesting possibly novel mechanisms at play. Here, we assess all available mitochondrial genomes in cephalopods using synteny and phylogenetic methods, and include for the first time comparisons between individuals within species. Like previous studies, we find independent evolution of replicate control regions in cuttlefish, oegospid squid, and loliginid squid, and duplicated genes in oegospid squid. Sequences were up to 100% identical at paralogous loci within an individual but divergent for orthologs between individuals. These findings highlight a tension of sequence convergence within but divergence between individuals, and raise as questions for future studies, how and where in the lifecycle and anatomy does convergence vs. divergence occur?