We should be a bit careful with what one means by "modulo arithmetic" here. If we are talking about arithmetic in Z/nZ (read "integers modulo n"), then the objects being acted upon are no longer integers, but equivalence classes of integers. That is, the set of all integers that satisfy the equivalence relation "~" where "a ~ b" means a - b = k*n for some integer k. For example, one of the equivalence classes in Z/3Z would be the set {..., -4, -1, 2, 5, ...}.
Since every element of the set is equivalent under that relation, we typically will choose a representative of that equivalence class, e.g., [2] for the previous class. If we view the "mod" or "modulo" operator to be a mapping from the integers to a particular representative of its equivalence class, there's no reason that negative values should be excluded. [-1] refers to the same equivalence class as [2], [32], and [-7]. The details of how integers are mapped to a chosen representative seem to vary from language to language, but modular arithmetic works all the same between them.