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Do I control the unsubscribe handling or does Mailgun?

It’s up to you.

To use Mailgun’s unsubscribe handling, you can include our unsubscribe variables %unsubscribe_url% (for the entire domain) and %tag_unsubscribe_url% (for just emails with this tag) and we will take care of the unsubscribe handling for you.

Take a look at our unsubscribe documentation for more information.