EDIT: My problem has been fixed or, more precisely, worked around, thanks to help in the comments below. Never figured out what exactly the issue was as it wasn't a problem with my Emacs configuration but with my Emacs installation (which I had originally done via Homebrew).
I am using Emacs 25.0.5 on OS X 10.10.5. For some reason, Emacs displays
‘ ’ and
“ ” only as
' ' and
" ", respectively. The behavior persists in
typopunct-mode (from the Typopunct package), although things like en- and em-dashes display correctly as distinct Unicode characters. The behavior also persists if I use a variable width typeface I am certain has glyphs for curly quotes.