I think they are most likely daft (and made an incorrect assumption) rather than Machiavellian. Don't go around all guns blazing - no point in starting a war with neighbours. Just be friendly and swallow down your irritation - explain to them about the deeds and that it is your private garden, and that unfortunately it seems that they weren't aware. This may end amicably with them apologising and feeling foolish about their error - at least try for that outcome?
If however they are not in the mood to listen / are adversarial and defiant, then you should take further action