This is a huge point of contention, which is why it must be agreed upon in the contract before the partnership commences (yes, there is a contract -- you'll be hearing more about it in a moment). The couple I worked with wanted twin boys, and luckily that's what they ended up with, because I said I wouldn't selectively reduce for any reason other than health risks to me or the babies. Since some of those risks can be caused by genetic abnormalities, I agreed to undergo genetic testing early in the pregnancy, which became quite literally a sticking point. They wanted to do an amniocentesis, which is a horrifying process involving jamming a needle into your belly and extracting amniotic fluid for testing (it also happens to present a higher risk for miscarriage and other issues). In the end, we agreed upon a less risky procedure.
And, as messy as those disagreements can be, now add in the fact that no state really enforces the surrogacy contracts and that paid surrogacy is illegal in many states. That means that, contract or no, parents can refuse to take custody of a "defective" baby under the "you birthed it, you bought it" statute, so you might end up going home with someone you didn't intend to, which rarely works out well. There have also been problems with surrogates who decided they wanted to keep the baby after it was born, and there's no legal way of stopping them. To prevent any of that awfulness from happening, they do the DNA test right there in the hospital to legally declare that it is the couple's genetic child.
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"For the last time, Mr. Povich, we're not hiring."