“והיה לאות על ידך ולזכרון בין עיניך” (י”ג,ט’)
There are four parshiyos in the teffilin. The מחבר (סימן ל”ד סעיף א’) brings the famous machlokes between Rashi and Rabbainu Tam in regard to the order of these parshiyos. Everyone agrees that they must be written in the order in which they are written in the Torah קדש, והיה כי יביאך, שמע, והיה אם שמע. Rashi holds that this is also the order in which they are put in the teffilin. In the Teffilin shel rosh the first parsha is put in the compartment all the way to the left of the one wearing it, followed by the next parsha. According to Rabbainu Tam the order is קדש, והיה כי יביאך, והיה אם שמע, שמע. However, as stated above שמע still has to be written before והיה אם שמע. According to Rabbainu Tam, when writing the Teffilin shel yad which is on one klaf, he first writes קדש, והיה כי יביאך then he leaves enough space for והיה אם שמוע, then he writes שמע and goes back to the empty space and writesוהיה אם שמע .
This is really already a machlokes between the Mechilta who holds like Rashi and the Yerushalmi (קדשים מובא בספר האשכול) who holds like Rabbainu Tam. The מחבר (סעיף ב’) says ומנהג העולם כרש”י והרמב”ם – the accepted practice is to do like the opinion of Rashi. This is also the opinion of the Gr”a.
The מחבר (סעיף ב’) writes that a ירא שמים should put on both pairs of teffilin. However in הלכה ג’ he says that only someone who is מחזק ומפורסם בחסידות – one who is well known to be pious in everything he does should also wear the teffilin of Rabbainu Tam. Even if one wants to put them on after davening he should not do so publicly for he is מיחזי כיוהרא – he is making himself look as if he is better than everyone else. However the ערוך השלחן writes that today when there is a widespread minhag of people putting on Rabbainu Tam teffilin, there is no longer a problem of מיחזי כיוהרא and anyone who wants can do so ותבוא עליו ברכה. The שערי תשובה as well brings from the ברכי יוסף that it is no longer considered מיחזי כיוהרא. Most people only put on after their wedding. The seforim give a reason for this. However there are those who put on even before their wedding.
Reb Chaim Volozhinor put on teffilin of Rabbainu Tam. He once asked his Rebbi, the Vilna Gaon, why he doesn’t put on teffilin of Rabbainu Tam. The Gaon asked Reb Chaim why he puts them on, to which he replied to beיוצא the ספק of both שיטות. The Gaon then said, if so, you should be חושש for all the other שיטות also and put on sixty four different pairs of teffilin! The מנהג of the חזון איש was not to put on teffilin of Rabbainu Tam. The חפץ חיים and the Steipler both said that one should try to put on Rabbainu Tam teffilin at least once in his lifetime. Rabbainu Tam teffilin are not worn on Chol Hamoed.
Prepared by R’ Avrohom Yehoshua Ziskind
 Sources:שו”ע ומשנה ברורה,אוצר פלאות התורה,פסקי תשובות,אוצר פלאות התורה