[Note: I’m going to start writing about technical fixes here. This is less for my readers, and more just to note certain tricky or non-intuitive fixes for myself so that I can look back and reference them. Do not feel obliged to read this.]
Today the language-switcher module on the YTG site stopped working. It would send the user to an incorrect page. Either the home page, or a page I had visited recently. The last part was the crucial hint. After turning cloudflare caching off, I remembered that I’d recently enabled caching within the Global Settings menu in Joomla.
Turned it off. Problem solved.