First of all, write down everything that has happened in a kind of list form with the date, as closely as you can remember it. Keep it short and factual, e.g. "Sat 14 Nov: allowed access for 2 hours for gutters to be cleaned. Dirt from gutters dropped onto walk way and left by contractor, required cleaning. Sat 21 Nov: neighbour called seeking permission for contractor access for..."
From now on write down what happens (and constantly watching you in your garden or entering and leaving your house counts as something happening).
You have a choice: contact the landlord giving a summary of what is going on and saying you do allow reasonable access but the neighbour is not being reasonable, or watch and wait, i.e. leave it unless you get some communication from the agent and the landlord that you have to answer.
You can always go to the police to complain she is harassing you. It may not be at a stage yet where you want to do that, but bear it in mind. If your neighbour starts to abuse you or interfere with your property, do it immediately. The fact that she is obsessive about getting the guttering done and refuses to admit it has been done recently is not a good sign, to be honest. Sounds very odd.
And finally, even when you do allow access you retain the right to have a say over how that access is used. Don't let any contractor get away with leaving a mess or being inconsiderate and blocking your door. If they don't comply with your requests, tell them to leave.
The fact that you are tenants not owners does not mean you don't have the right to control your environment: you do.