The special relationship between Emily Grierson and the town is actually a rather warped, unhealthy one. Emily represents what for many people in the town is a vanished golden age, an age of fine Southern grace, order, and social stability. Among other things, this means that Emily gets a pass for her rudeness and eccentricity. The town's mayor even exempts her from paying taxes.
Unfortunately, this "special" relationship means that people look the other way—or rather turn their noses the other way—when horrible smells start coming from the old Grierson residence. It's only after Emily's death that the true source of this stench is shockingly revealed. And it's only then that the strange spell that Emily and her family had cast over the town for so long is finally broken.