The main basis for putting on Rabeinu Tam Tefilin is, as you probably guessed, the opinion of Rabeinu Tam in Menachos (34b, DH "viha'Korei"), which many say is also the opinion of the Zohar. Some Sefardim wear both at the same time, as the Shulchan Aruch (O.C. 34:2) seems to codify that this is the proper conduct of a Yarei Shamayim.

Some Sefardim only put on one pair at a time, in accordance with the Shulchan Aruch's statement (ibid.) that if one is unsure whether or not he will put both Tefilin in a place where he will be Yotzei the Mitzva of Tefilin he should only wear one at a time (Rashi first and Rabeinu Tam second).

All the best,

Yaakov Montrose