EDIT: Updated with the actual default command.
I think this may not be the default; it doesn't work for me.
While speaking of the character picker, does anyone know a way to get less broken search in the picker? For example, I frequently find myself looking for math italic characters in Unicode. Despite the description being, say, "MATHEMATICAL ITALIC SMALL A", it doesn't come up when I search for 'small'. In this case searching for 'mathematical' or 'italic' works, but there are other cases where I just have to guess search terms randomly (or go look up Unicode tables elsewhere).
Indeed, I thought that might be what you meant … but that opens the Finder search window for me, so I think that that shortcut also is not universal. Probably some of these differ depending on whether you've got a fresh install or an updated system that may carry along shortcuts from an older release.
EDIT: adsche (https://news.ycombinator.com/item?id=24092377) suggests ⌘⌃␣, and that works for me.
Completely agree about the search. E.g., I sometimes want the cmd sign (⌘), which is named "place of interest". Sure, that makes sense from unicode and in other contexts but I would expect to find it as cmd or command in macos.
There are a LOT more sets of characters available in the picker, hit the gear menu in the upper left and select "Customize List".
Tried in both Chrome and TextEdit.