Those things aren't at near-constant density, though. And although the article shows that icebergs do flip, the fact that the Smithsonian wrote an excited piece

Those things aren't at near-constant density, though. And although the article shows that icebergs do flip, the fact that the Smithsonian wrote an excited piece about it suggests that it doesn't happen too frequently. And it sounds like that happens because pieces break off or melt. The examples in the image search aren't just kinda unstable; they're completely implausible.