I'd go with dzug - I had a similar thing where a neighbour wanted a fence, and just paid for it. Unless you jointly pay, whoever did pay has say over it and it remains their property, so unless agreed, you cannot just tear it down or put up another one, or even attach anything to it unless you have permission.
Regarding the thrown cuttings you mentioned, were they YOUR bushes which had simply grown over? If so, your neighbour was simply complying with the law, in that he is entitled to trim anything that encroaches on his garden, but, because it is your property, has to give that back to you.