Learn Italian: Grazie per l'invito! Figurati! - "Thanks for the invitation!" "Don't mention it!"

It comes from the reflexive verb figurarsi, which literally means "to imagine" or "to figure to oneself." As a fixed expression, Italians use it to mean "Don't mention it," "No worries," or "It's nothing." You are essentially telling the other person to imagine that their request or thanks is no burden at all.

What grammatical form is figurati?

It is the informal imperative, or command, form for tu. It combines the conjugated verb figura with the reflexive pronoun ti attached to the very end.

How would I say "Don't mention it" to someone I address formally with Lei?

You would use the formal imperative form: si figuri. Notice how the reflexive pronoun si comes before the verb in the formal command, which is the opposite of the informal figurati where it attaches to the end.

Could I say grazie dell'invito instead of grazie per l'invito?

Yes! When thanking someone for a noun like an invitation, you can use either per or di. Both grazie per l'invito and grazie dell'invito are perfectly natural and mean the exact same thing.