Whew, I fixed it!
In Keychain Access (the OS X utility), find and highlight the offending DigiCert Root CA certificate. Open its info, open the "Trust" area, and change "Use System Defaults" to "Always Trust".
That should take care of signing in at auth.opera.com. Do the same thing for the offending Symantec certificate to get onto Twitter.
I don't think it matters with the above steps done first, but if they don't work, go to the /var/db/crls folder and delete everything there, or at least crlcache.db and ocspcache.db. Restarting the system might be necessary, too.