You can automatically crosspost between Mastodon and Twitter in either direction using various 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.
Unfortunately as of February 2023, Twitter’s recent API shutdown has caused many free crossposting services to close down or announce they’re closing down. However, professional crossposting services such as Buffer ⧉ are still working and now support Mastodon.