Rashi and Rabbenu Tam — the order of the four

Four scrolls, four compartments, and the Talmud does not settle beyond dispute which scroll goes in which compartment. So the medieval authorities did, twice, differently — and the disagreement is between a grandfather and his grandson.

Rashi (d. 1105): the order runs in the order the passages stand in the Torah — Exodus 13:1–10, Exodus 13:11–16, Deuteronomy 6:4–9, Deuteronomy 11:13–21.
Rabbenu Tam, his grandson (d. 1171): the last two are reversed, so that Deuteronomy 6 sits on the outside rather than inside — the Shema at the edge of the box, not buried in the middle of it.

Nothing visible distinguishes the two. Two boxes made to the two opinions are, from outside, the same object; the difference is entirely in what order four hidden scrolls were laid in, and it cannot be checked without destroying the box. Practically the whole world follows Rashi. A minority — much of the Hasidic world, and many Sephardi communities — wear a second pair to Rabbenu Tam's order after the first, on the grounds that one of the two is right and this way you have certainly done it.

Layer: this is halakhic argument, twelfth century, and it is argued from the wording of Menachot and from the shape of the verses. It is not a mystical dispute, although the Kabbalists later attached readings to both positions — the Ari is reported to have worn both, which is itself part of why the practice spread.