We're looking for vertical to horizontal switching. Not just examples of vertical writing.

Almost all extant vertically-written languages are more commonly seen horizontally on the web. Vertical Japanese on the web (from my American understanding) is a design choice, and is notably absent from mainstays like https://www.yahoo.co.jp/

That is to say, it is unlikely that a site's language switcher would opt for top-to-bottom writing.

Edit: the Mongolian president's site does have an English version! ...but it's a completely different site :( https://president.mn/en/. Still, this is the closest to a use case for logical properties I've seen, so kudos.