Oh my, MY neighbor does the SAME thing!
Since the day we've moved in here.. they mow the back part of our lawn. They planted a huge garden RIGHT on the edge of our yard (we have three boys, who play ball.. they might want to move the garden!). They have 2 acres.. we have a 1/2 acre.. really.. RIGHT on the edge, wasn't necessary. They hauled off a piles of sticks that I had in my yard. They've trimmed our trees. (that one was the final thing that got DH over talking to them.. asking if they understood where the property line was).
They owner of the land next to us drives through our yard.. because she says it's her yard. She comes over and asks what we're doing, if we're in the "questionable" area. She is nice.. don't get me wrong. She said the boys can play on any of her yard, just don't leave toys laying around. I do greatly appreciate that. So, I dropped the whole thing.
This year.. I just told DH.. they can mow it if they want. Save me some work. We're putting the house on the market this spring anyway.
I would say take a soil sample to your local extension office. They should be able to tell, at a fairly minimal cost.