Mrs O, I have a nearly 8 year old DSS who has been (at times) somewhat the same (and like your DS, he is 50:50 shared care between us and his mum).
I must say that at the moment though, out of all my three step-kids, he is the one who is least trying! He actually DOES remember to hang up his bath towel, hasn't lied to us for ages, generally does what he is asked (although it might take a few times asking!).
A while back he was quite challenging, acting up at school, inapporpriately rough play, disruptive in class etc and the teacher worked on a program of him thinking about his actions, a star chart with rewards etc and it seemed to really pay off with a great report at the end of last year, and this year, all is going pretty well. At home, we did have a few episodes of blatant lying and strange/rude/challenging behaviour, after one particular episode involving the small children next door, DH & I took him aside and had a very serious talk with him - I still remember the look on his face when we took him into the office, I think he thought we were going to be really mad with him but we were very calm and quiet, asked him for an explanation, which came out sounding like total BS, we told him we thought he was lying, and quietly kept working on getting an explanation and told him he would have to go in next door and apologise to both the children and their parents, which he did (with my DH). I think that was a bit of a turning point, it made him see the seriousness of a situation, the consequences of making bad choices and of compounding it by lying.
I tell this story just by way of example, if you think your son is up to some serious but quiet, measured talking-too, including having to face up with apologies to anyone he is affecting by his actions, then maybe he might respond? But you would be best placed to know how he deals with things given his Aspergers.