You can automatically crosspost between Mastodon and Twitter in either direction using third party services. Crossposting means when you post on Mastodon the same content will also be posted on Twitter (or vice versa).
This can be controversial if you’re just mirroring your Twitter account to Mastodon without interacting on Mastodon at all. Some servers specifically ban such “zombie accounts”.
Going in the other direction, if you’re active on Mastodon and mirroring it to Twitter, there’s no controversy at all.
Which crossposting services are available?
Professional paid crossposting services such as Buffer ⧉ and Publer ⧉ are still working.
However, most free crossposting services have shut down due to Elon Musk’s shutdown of Twitter’s free API.