It's important to remember that Minneapolis is actually the merging of two cities, Minneapolis and St. Anthony. St. Anthony is everything east of the river (so the NE and SE quadrants). At one point in time St. Anthony was larger than Minneapolis. There was a large competition between the two cities because of their flour mills, Pillsbury on the St. Anthony side, and Washburn-Crosby (modern day General Mills) on the Mpls side.

The cities merged because of competition with St. Paul, which had better rail access and was further down the river (didn't have to deal with the waterfall in Mpls).

So the original grids of both cities were created independently, however, once the cities merged they renamed almost all of the streets in both cities to be what is the modern day naming. This was mainly because there were many duplicate names on either side of the river. If they hadn't done this the street naming would be utterly confusing.

So the answer to your question is really "it's complicated".