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Rabbi Zwecker רב צווקר
There is much discussion among the Rishonim (early authorities) about the order in which the biblical passages should be written in the arm and head Tefillin. The rabbis most famous for this dispute were Rashi and his grandson Rabbenu Tam.
Rashi held that the order is: Kadesh Li (Shemos13:1-10), VeHaya Ki Yeviehcha (Shemos 13:11-16), Shema (Devarim 6:4-9), VeHaya Im Shemoa (Devarim 11:13-21).
Rabbenu Tam held that the last two passages were switched (thus making the following order Kadesh Li, VeHaya Ki Yeviehcha, VeHaya Im Shemoa, Shema).
The custom of the velt is to follow Rashi's opinion.[1]
Rambam originally held like Rabbenu Tam but then retracted and ruled like Rashi after hearing of the discovery that Rav Hai Gaon's Tefillin were like Rashi.[2]
Two other orders exist, known as Shimusha Rabba Tefillin[3] and Raavad[4] which are the reverse of Rashi and Rabbenu Tam Tefillin, respectively.[5]
Some say that they only differ in the Shel Rosh, but they are exactly the same as Rashi/Rabbenu Tam in the Shel Yad.[6]
How Shimusha Rabba Tefillin are Made:
There are a few defining details about Shimusha Rabba Tefillin, including the taggin (serifs), reversed order of the parshios, and etzbayim by etzbayim size.
According to some, they are essentially Rashi Tefillin made to size, whereas others think the order of parshiot must also be reversed.[7]
When Shimusha Rabba Should Be Worn?
The Arizal used to wear Rabbenu Tam Tefillin by Mincha, but, when he made a pair of Shimusha Rabba Tefillin, he began wearing that when davening Mincha instead.[8]
Even though it is uncommon, Mekubalim follow the minhag and practices of the Arizal, they wear Shimusha Rabba by Minchah with a Berachah.[9]
Some say one should not wear Shimusha Rabba by Mincha with a Tzibbur if nobody else does, because that would be Yuhara - arrogance and false pride, unless the person is well known for his piety. So, unless others are aslo wearing them, one should put them on privately.[10] Others disagree.[11]
One does not wear them on Erev Shabbos or Erev Yom Tov, but he does on Rosh Chodesh.[12]
Some recommend wearing them after Shacharis.[13]
Who Should Wear Shimusha Rabba?
One who lives an outstandingly pure life may wear Shimusha Rabba, perhaps after forty years of age.[14]
Sources
1Shulchan Aruch O.C. 34:1, Halacha Brurah 34:1, Kitzur Shulchan Aruch of Rav Rephael Baruch Toledano, siman 10, miscellaneous laws (of tefillin), seif 2.
Shut Min HaShamayim 3 says that the debate between Rashi and Rabbeinu Tam is also a debate in Heaven. See Rav Reuven Margaliot's notes ad loc. where he presents an exposition demonstrating how the debate between Rashi and Rabbeinu Tam precedes them to the times of the Tannaim and the Rishonim are only debating which view is accepted as Halacha. According to the Arizal, there's no machloket: both Rashi and Rabbeinu Tam are true. (Or LeTzion 2:3:8) There are various approaches to understanding how this debate began. See Rav Menachem Mendel Kasher's essay in Torah Shleimah vol. 12 Miluim page 271. Rav Kasher postulates that it's impossible for there to have been a break in the Mesorah due to persecution or the like and that in his opinion perhaps the order is not important on a Torah level. Only on a rabbinic level, the Chachamim sought to universalize the order of the Parshiot, similar to the Ritva's point about the Teruah on Rosh HaShanah 33a, and the debate is about what the decision was, due to the many leftover pairs of Tefillin from before the decision was made.
2 Shu"t HaRambam Siman 7, Kesef Mishneh (Hilchot Tefillin 3:5)
3 as presented by the Rosh (Halachot Ketanot, Hilchot Tefillin 5), which is the opposite of the Raavad's version of Shimusha Rabba
4 Temim Deim Siman 79. The Raavad's version of Shimusha Rabba read the opposite of the Rosh's.
5 See Shu"t HaRama MiFano 107
6 Piskei Teshuvot 34:1
7 Shaarei Halacha uMinhag Orach Chaim 1:31
8 Kaf HaChaim (Orach Chaim 37:10)
9 Or Letzion 2:3:12, Yalkut Yosef (Orach Chaim 37:4). They both argue that since the Shimusha Rabba are essentially Rashi Tefillin, one should recite a Beracha according to Peshat, even though the Rama Mifano 107, Chida (Machzik Beracha Orach Chaim 34:4), and Yaskil Avdi (vol. 8, Deah VeHaskel 6:8) argue that one should not recite a beracha because the Tefillin are so holy. See Halichot Olam vol. 1 pg 25
10 Yalkut Yosef (Orach Chaim 37:5)
11 Or Letzion 2:3:12
12 Yalkut Yosef (Orach Chaim 34:16, 37:4)
13 HaYom Yom 19 Menachem Av, Shaarei Halacha uMinhag Orach Chaim 1:31 See there for the step by step guide to wearing all four pairs every morning.
14 Shaarei Halacha uMinhag Orach Chaim 1:31
The Lubavitcher Rebbe zt"l relates that "In order to put on four pairs of tefillin, one's body has to be exceptionally pure. For with regard to duties obligated by the Torah, such as the tefillin of Rashi and Rabbenu Tam, since "The Torah was not given to the ministering angels," the Torah takes the responsibility upon itself, so to speak. By contrast, with regard to things observed by very few people (the Torah not having obliged everyone to observe them), the individual responsibility is greater. Besides, the very nature of the tefillin of Shimusha Rabba and Ra'avad requires that the body be exceptionally pure." And he adds in a footnote "See Igrot Harames, sec. 5: “The Tefillin of Shimusha Rabba… require an untainted body, and pure and clean thoughts.” R. Menachem Azariah of Fano writes in his Responsa that “these tefillin plumb such deep mysteries that permission was barely granted to use them.” The author of Emek Hamelech writes in Sha’ar Kiryat Arba. 68: “These tefillin, relate to [the transcendent level of Divinity known as] Arich, and which are arranged according to the teachings of the author of Shimusha Rabba."