it’s a machlokes that dates to way earlier than the rishonim, In an interview published in Ami a while ago, a YU professor (I forget who) said that archaeologists have found 2 complete pairs of tefillin in the bar kochva caves that they could actually open without damaging them. One was ordered according to Rashi, the other according to Rabbeinu Tam.
Also, if DY (and the rema http://hebrewbooks.org/pdfpager.aspx?req=14170&st=&pgnum=80)
are right that the order is meakev, then wearing 2 pairs isnt a halachic issue at all. if it was really about a chashash of being yotzei bedieved, then you would make a beracha on both pairs and wear both for kerias shema.