When they rolled it out, it was SMS-only. It still doesn't use TOTP - they've baked something into their mobile app that allows it to function as the second factor, rather than just supporting the same 2FA scheme that everyone else does.
Since they don't use TOTP it's not possible to use their two-factor authentication with applications such as Google Authenticator or Authy. What's more, I don't think it even works with third-party Twitter clients (Correct me if I'm wrong), so you have to use Twitter's own apps.
It does indeed work with third-party apps using Google-style "app specific passwords". But I didn't know that it didn't support TOTP since I only use SMS authentication. I can see that as being a lame move, though.