Rabbenu Tam Tefillin
Jewish men wear at least one pair of Tefillin every weekday. Both Tefillin Shel Rosh and Shel Yad contain four biblical passages (parshiot), the order of which is disputed primarily by Rashi and his grandson Rabbenu Tam, resulting in "Rashi Tefillin" and "Rabbenu Tam Tefillin," each one written with the parshiot according to one of the two views. Due to unresolved nature of the debate, the pious practice developed to wear each pair of Tefillin every day, in order to fulfill both opinions. There are also additional types of Tefillin, known as "Shimusha Rabba Tefillin" and "Ra'avad Tefillin that are worn by people of outstandingly pious nature.
Arrangement of Paragraphs (Seder HaParshiot)
|Rashi||Kadesh Li||Vehaya Ki||Shema||Vehaya Im|
|Rabbeinu Tam||Kadesh Li||Vehaya Ki||Vehaya Im||Shema|
|Shimusha Rabba||Vehaya Im||Shema||Vehaya Ki||Kadesh Li|
|Ra'avad||Shema||Vehaya Im||Vehaya Ki||Kadesh Li|
- 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 (Shemot 13:1-10), VeHaya Ki Yeviehcha (Shemot 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 world is to follow Rashi's opinion. 
- Shimusha Rabba Tefillin
Who Should Wear Rabbenu Tam Tefillin?
- Pious individuals have the practice of wearing both Tefillin of Rashi and Rabbenu Tam, with no bracha when putting on the Rabbenu Tam. 
- Unmarried individuals should not wear Tefillin of Rabbenu Tam unless they know that they are capable of having pure thoughts for as long as they are wearing Tefillin. 
- No beracha is recited when putting on tefillin of Rabbenu Tam. Nevertheless, one should not speak between the shel yad and the shel rosh. 
- If one mistakenly put on Rabbenu Tam first, some poskim say to recite a bracha when putting on the [Rashi]] Tefillin.
- The Rashi and Rabbenu Tam Tefillin should be placed in separate bags. 
- Some say that a person should wear Rabbenu Tam tefillin at least once in their life.
How Shimusha Rabba Tefillin are Made
- There are a few defining details about Shimusha Rabba Tefillin, including the taggin (serifs), reversed order of the parshiot, and etzbayim by etzbaim size. According to some, they are essentially Rashi Tefillin made to size, whereas others think the order of parshiot must also be reversed.
When Shimusha Rabba Should Be Worn
- The Arizal used to wear Rabbenu Tam Tefillin at Mincha, but, when he made a pair of Shimusha Rabba Tefillin, he began wearing that at Mincha instead.
- Even though it is uncommon, Sephardim follow the practice of the Arizal, so those who wear Shimusha Rabba do so at Mincha with a Beracha.
- Some say one should not wear Shimusha Rabba at Mincha with a Tzibbur if nobody else does, because that would be Yuhara unless the person is well known for his piety. So, unless others are aslo wearing them, one should put them on privately. Others disagree.
- One does not wear them on Erev Shabbat or Erev Yom Tov, but he does on Rosh Chodesh.
- Some recommend wearing them after Shacharit.
Who should Wear Shimusha Rabba
- One who lives an outstandingly pure life may wear Shimusha Rabba, perhaps after forty years of age.
- Ten Minute Halacha: Rabeinu Tam Tefillin by Rabbi Aryeh Lebowitz
- Order of Parshiyos in Tfillin: Rashi and Rabbeinu Tam by Rabbi Zvi Sobolofsky
- Tefillin: Shimusha Rabba and Ra’avad by Rabbi Ari Enkin
- Shulchan Aruch 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)
- Kitzur Shulchan Aruch of Rav Rephael Baruch Toledano, siman 10, the order of the donning, seif 2. Shulchan Aruch 34:2-3 rules that that the especially pious should wear both sets but adds that only one commonly known to be especially pious may do so, so as not to appear as arrogant in one's piety. However, Halacha Brurah (Rabbi David Yosef, 34:1) writes that since it has become the practice of many common people to wear both Tefillin of Rashi and Rabbenu Tam it is not considered an issue of arrogance to wear both pairs of Tefillin. Aruch Hashulchan 34:5 agrees. Sh"t Yabia Omer OC 3:4 writes that if one has this stringency he should remove the Rashi and place the Rabbeinu Tam before the kedusha of "uva litzion".
- Halacha Brurah 34:2. Chacham Ben-zion Abba Shaul Sh"t Or Litzion 2:13:11 also writes that Sephardim should only wear Rabbenu Tam Tefillin after marriage. Divrei Yatziv 42 and Mishneh Halachot 6:12 agree
- Sh"t Yabia Omer 3:3
- Ben Ish Chai Vayera 24. Rav Ovadia Yosef in Halichot Olam volume 1 page 26 says that this is only if the person has prayed something in between but if he remembers immediately no bracha should be recited.
- Kitzur Shulchan Aruch of Rav Rephael Baruch Toledano, siman 10, the order of the donning, seif 5
- Aruch Hashulchan 34:5 writes that since ti si common to wear them in some places it isn't yuhara and praiseworthy is one who wears them. Igrot Moshe OC 4:9 writes that since he had particular concerns of whether they were kosher he didn't wear rabbenu tam tefillin. Dirshu 34:24 cites Dinim Vhanhagot 3:15 of the Chazon Ish that he didn't wear Rabbenu Tam tefillin. However, the Chafetz Chaim and the Steipler (Meir Oz tefillin p. 481) held that a person should wear them once in his life.
- Shaarei Halacha uMinhag Orach Chaim 1:31
- Kaf HaChaim (Orach Chaim 37:10)
- 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
- Yalkut Yosef (Orach Chaim 37:5)
- Or Letzion 2:3:12
- Yalkut Yosef (Orach Chaim 37:4)
- HaYom Yom 19 Menachem AvShaarei Halacha uMinhag Orach Chaim 1:31 See there and HaYom Yom, Menachem Av 19 for the step by step guide to wearing all fur pairs every morning.
- Shaarei Halacha uMinhag Orach Chaim 1:31