My neighbor wants to replace her entire fence, including a panel between our two properties. It's not because it needed repair or anything. She just wants to have all three sides of her fence match visually. But then my three fence panels won't match! On top of that, she sent a letter demanding that I pay one-third of the cost of her replacement, since I "benefit" from her new fence. What should I do?
A considerate neighbor in this situation would, of course, realize the impact of her decision and consult you. But not all neighbors are considerate people. Your neighbor might not realize the effect that this will have on your house, resulting in three non-matching panels. Or, your neighbor might simply not care about the effect on your home.
Either way, you need to initiate a conversation. Your initial leverage is that there is no requirement that you contribute financially to pay the one-third share she is demanding, given that this is not a repair situation. It sounds as if this is what's known legally as a "boundary fence," located on the line between two properties and used by both owners. If so, both owners are responsible for keeping the fence in good repair; which does not mean having to pay up every time one owner wants to swap the fence for a different style.
You need not take a hostile tone, at least to begin. You might say something like, "I would be glad to contribute a one-third share. But since I would like all three sides of my fence to match, let's try to find a style that will work for both of us." This makes it clear that you're not unreasonable or unwilling to contribute, but you're also not a pushover. You have no intention of having a non-matching fence, and paying for the privilege!
Also consider getting in touch with your neighborhood or homeowners' association (HOA), if you live in a planned community that is governed by one. Many neighborhood associations, particularly in small suburban communities, have specific regulations regarding the design of common elements like fences. Their aim is to maintain an aesthetic uniformity from house to house, so that a chain-link fence isn't followed by a white picket fence and then followed by a post-and-rail wooden fence. Perhaps your neighborhood association has a number of styles from which you and your neighbor can pick.
If these approaches do not work, your fallback option is to refuse to pay anything, and then add a new panel on your own property that matches your initial two, such that there would be two fence panels between your home and your neighbor's. This is not uncommon in suburban areas. But it is also highly inefficient, and will cost both of you more than simply agreeing on a fence design. It is in both of your interests to find an aesthetic that you agree on, and share part of the cost.