Following fertilization in mammals, it is generally accepted that totipotent cells are exclusive to the zygote and to each of the two blastomeres originating from the first mitotic division. This model of totipotency was inferred from a minority of cases in which blastomeres produced monozygotic twins in mice. Was this due to experimental limitation or biological constraint? Here we removed experimental obstacles and achieved reliable quantification of the prevalence of dual totipotency among mouse two-cell stage blastomeres. We separated the blastomeres of 1,252 two-cell embryos, preserving 1,210 of the pairs. Two classes of monozygotic twins became apparent at the blastocyst stage: 27% formed a functional epiblast in both members (concordant), and 73% did so in only one member of the pair (discordant) - a partition that proved insensitive to oocyte quality, sperm-entry point, culture environment and pattern of cleavage. In intact two-cell embryos, the ability of sister blastomeres to generate epiblast was also skewed. Class discovery clustering of the individual blastomeres' and blastocysts' transcriptomes points to an innate origin of concordance and discordance rather than developmental acquisition. Our data place constraints on the commonly accepted idea that totipotency is allocated equally between the two-cell stage blastomeres in mice.