You can crosspost between Mastodon and Twitter in either direction using various third party sites.
When crossposting is switched on, it means when you post on Mastodon it 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".
However, if you're mainly active on Mastodon and mirroring it to Twitter there's no controversy at all.
@feditips @Lucseleventje @crossposter @moaparty @renatolond @flancian You have to set up the cross-poster on desktop (not sure if the initial set up works on mobile). Once you post via mobile, it works fine regardless of the direction the cross-poster is set. We use Tusky and we have no issues with posts showing up on Twitter.
Yes, once you set up bridging (https://moa.party *should* work on mobile too for setup, but I think I haven't tested it lately; please let me know if it doesn't!) you should be able to post and read from any client; cross-posting happens in the moa.party backend using your stored credentials (after signing in), so it's independent from your client. Moa keeps track of which messages it has x-posted, etc.
@feditips Why is this controversial? Isn't the point to allow users the freedom to consume content where they want?
When Mastodon first got popular, some people from Twitter set up "zombie accounts" on here where they pretended to be interacting but weren't really, they were just blindly reposting whatever they did on Twitter.
It became difficult to know who was a real user on here and who was a "zombie", so some servers started banning such users.