How can we promote engagement?

I suppose he eats.

Ask him to make his own food.  Or go out and do his own shopping.

Yes, he actually does this.  A small but vital area of independence.

He pops to the local co-op to top up what he gets in our weekly online shop.  Generally he cooks his own meals either during the night or when we're not around, although (as an act of caring) I do sometimes leave out special prepared meals for him that I know he'll like.  He won't come into the kitchen if anyone else is there and, if, for example, I'm coming up the stairs just as he's coming out of his room, he hastily goes back in there until he's sure I've gone.  :(  He also does his own washing, again during the night when there's nobody around.  

His meals are always taken back to his room, even though we often invite him to eat with us or, say, watch a film or share a takeaway.  He occasionally asks me a question about food or shopping but is generally halfway back up the stairs, so's to indicate he doesn't want a long answer.  If I try to initiate a conversation, the door is closed and he either doesn't respond, as if conversation is actually painful to him, or he answers monosyllabically.  I keep things warm and friendly and will continue to invite him, but it feels as though he sees us as the enemy or else for some other reason needs to shut us out.  

He has no friends (all eventually stopped coming round when he refused to go out), no courses or classes, no job or any other activities outside the home. 

I'm also concerned that his benefits will be discontinued as we have no recent corroborative evidence.  He ignores all letters from dentists and GPs, has very poor self care (i.e. very mucky, in spite of washing his clothes as he never washes himself) and doesn't clean his teeth. 

 I'm finding it very hard to live like this.   How can I encourage more interaction or activity?   Services won't help because, of course, he would have to request it himself.   

In some ways your son reminds me of myself around that age.  He can change.

The big difference is my parents were worse than me.

When I was 18 I registered myself with a local NHS dentist.  Before that dentists were not talked about.

As I did more shopping, they did less.

As long as I was living at home as a hermit.  My mother was content and my father just pretended that I didn't exist.

Yes, this would be the pattern I'd expect to see as we go through life and develop skills and coping strategies.  However, even given a major setback like a breakdown or bereavement, I would still expect to see a gradual "self righting" tendency.  I can't be sure that this isn't happening inside my son's mind so we're holding things steady and stable, but from what I can see we seem to be trapped in a statemate situation.  

Yes, this would be the pattern I'd expect to see as we go through life and develop skills and coping strategies.  However, even given a major setback like a breakdown or bereavement, I would still expect to see a gradual "self righting" tendency.  I can't be sure that this isn't happening inside my son's mind so we're holding things steady and stable, but from what I can see we seem to be trapped in a statemate situation.