One of the funniest moments in recent State of the Union history was back in 2006 when George W. Bush asked Congress "to pass legislation to prohibit the most egregious abuses of medical research: human cloning in all its forms; creating or implanting embryos for experiments; creating human-animal hybrids; and buying, selling or patenting human embryos."
That's right. No mermaids in Bush's America. And no Pigman either. But congress failed to act on the idea. It most recently came up in 2009, when for some reason Sam Brownback introduced a bill to ban human-animal hybrids that never made it out of the Senate Judiciary Committee. After that, everyone seems to have forgotten about it entirely. So as best I know, there's nothing legally stopping you from creating a centaur in your basement.
But as a sign of how much times have changed, here's the first thing that came up when I searched the 2006 SOTU for "hybrids":
We must also change how we power our automobiles.
We will increase our research in better batteries for hybrid and electric cars and in pollution-free cars that run on hydrogen.
Electric cars! Presumably the 2006 iteration of the conservative movement is thrilled that Tesla is an emerging American business and innovation success story, even if the 2014 iteration is stuck with a kind of paranoid lust for gas guzzlers.