All of this is assuming that the twins are identical – meaning they have the same DNA. Fraternal twins don’t share DNA and are no more or less alike genetically than any other brothers or sisters. Though there are some differences in DNA between twins due to mutations and other factors, let’s just ignore them.
Then what you have is two people with identical DNA, making babies with two other people with identical DNA. And that’s exactly the situation that happens if one set of parents make two babies – the same sets of DNA, reproducing twice. And we aren’t identical to our brothers and sisters – we just share some traits with them.
That’s because the DNA that ends up in an egg or sperm cell is a basically random half of the parent’s full genome – the egg and sperm each have half, and meet up to form a whole. And those halves are different between every single egg and sperm cell. I inherited my dad’s dark hair, so did my brother, but my sister is a redhead – she got a different set of hair color genes than we did. I have my mom’s big nose, my brother and sister both have my dad’s smaller nose – different sets of nose genes, coming from the same source.
So a set of babies born from two sets of identical twins would be, genetically, just like brothers and sisters. They’d share half of their DNA on average. So they’d be cousins to each other, but genetically closer than most of us are with our cousins.
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