As a conjunction, neither often teams with nor: "Neither a borrower nor a lender be." The rule many learned in fifth grade was, "Neither … nor, either … or, but never neither … or."
Neither … or is another once-unthinkable faux pas gaining momentum among people who ought to know better. A political adviser's resignation letter read, "This position is not a fit for me, neither personally or professionally." (Make it "either.")
A big-city newspaper editor wrote, "I neither commissioned or approved it." Even editors need editors.