I think the confusion regarding following the letter of the law versus the spirit of the law stems from the fact that Orthodox Jews practice the laws laid out both in the Old Testament, as well as in the Oral Law. Although the Oral Law was not written down until the Talmud, it was considered to be the word of God passed down through Moses throughout the generations. As such, it is central to the practice of Judaism in combination with the Written Law.
The issue of letter/spirit of the law is one of semantics. To Orthodox Jews, the Oral Law is considered the letter of the law. The rub is that the Law is subject to interpretation, as evidenced by the wide variety of opinions contained in the Talmud (along with the wide variety of rabbinical opinions regarding women’s hair covering).