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It's obviously somewhat circular. :)
It's not circular, it's a line! :-)
To take your arguments for (It should always be 'En dashes' and 'Em dashes') in turn.
Because the reference letters are uppercase
The name 'em' was in reference to the block size of the uppercase letter M, but that's no reason for it to be uppercase itself. If it was about the shape, like T-shirt or Q-tip, we'd maybe be in agreement.
because it's a name
Well, it's a noun, but it's not a proper noun, so it doesn't need a capital for that reason.
and to distinguish it from the measurement unit.
It is the measurement unit (the em of pica is a third of an inch or something) so doesn't need distinguishing from itself.
Feel like I may have lost the room long ago at this point, but have attached some screengrabs from the OED on the usages of 'em' for fellow nerds … en-joy!
Yes. "Coöperate" is nuts to me because the second o represents a perfectly normal English sound and a hyphen would work fine, if anything is needed at all. The broader point is that people and institutions love to make absolutist pronouncements about correct usage but they are rarely justified, and often only make things more effortful or jarring to the reader. DJing is easier to read than "D.J.-ing" is my point.
Ha! Now fixed!
Why? It has always been lowercase 'em' and 'en' in British English usage, used to signify the measurement of the type block 'N' and 'M'. Written as 'en' and 'em' to avoid confusion among printers between discussions about type measure and about actual letters. US or International English may differ idk.
I want to be a lion tamer!