He banged shut [the door].
He banged [the door] shut.
Both are correct.
Here is a very beautiful line consisting of similar construction from contemporary fiction genre:
The next moment she was grateful for the wind that had left Nelson Street and blown through the cane fields, banging shut the kitchen windows, so hard that they sprang open and struck the walls again, making their own echoes.
FROM - Bruised hibiscus by Elizabeth Nunez
He tore open [the parcel].
He tore [the parcel] open.
Here like the previous sentences both are correct.
In these cases shut and open are PARTICLES. Such Adjectival Particles are found in both idiomatic and non-idiomatic constructions.
Consider the following sentence -
He made [the boy] angry.
Here the adjective angry is not a Particle like open or shut as we saw in other examples earlier. And hence we can't say:
He made angry [the boy]. {INCORRECT]
Here angry is a Predicative Complement, and it is an object oriented Predicative Complement. And the predicand is the boy.
Just notice the word order: VERB + OBJECT + ADJECTIVE
but never 'VERB + ADJECTIVE + OBJEC'T.
But when it is a Particle, both word order is possible. While Prepositional Particle are found in abundance, adjectival Particles are very limited.