Where do the different methods of wrapping tefillin come from? What are the sources for how to wrap? I wrap in the following style. Who instituted this style and why? http://www.shemayisrael.co.il/tefillin/sefard.htm Is there a contradiction if I wrap outwardly, as in the picture, with Ktav Ha’Ari Tefillin, but use an Ashkenaz Siddur?

ב"ה Shalom The answer is similar to the one given in regard to your question of being consistent with the customs in regard to the writing of the Tefillin and the way the Tefillin straps are wound. The illustration in the link you posted, shows putting on the Tefillin outwardly. This is consistent with the Sepahrdic and Chassidic custom. The two ways of winding the tefillin are mentioned in the Bet Yosef of hilchot Tefillin. (Tur Orach Chaim 27) The direction the Tefillin are wound stems basically from the way the knot of the "yud" is tied on the Shel Yad. In the Bet Yosef, he quoted the Mahari Ben Chaviv who verbally describes the way the knot is made. The intention of each custom is to make the yud as close as possible to the "Bayit" of the Tefillin. The Rema who forms the basis for the Askenaz minhag says to wind the Tefillin inwardly which is opposed to the Sephardic and Chassidic minhag. All the best.