The original 7th was played over the ground that is the 12th fairway to the area of green we think of as the 11th. The 11th has not altered position and was that way even in the reversed original course. The reversed 12th is the same 12th, not to the area we know today as the 6th. 8, 9 and 10 were the same also.
I must admit, it had never occurred to me to think when the crossover was added to the course. Adrian's quote is the closest to explaining it, but doesn't quite go far enough.
Originally, the whole course was single-file and not double-wide as we know it today, and there are some distinct fairways now that barely existed then. #12 fairway was the way to #7 / 11 green back then, and most of #6 fairway and green was likely not there at all. [This makes me think twice about which others of today's holes were not part of the original nine out and nine back.]
So, the crossover happened when the course was widened to what we know today. They didn't want to give up #11 as we know it, and they didn't want play coming the other way to kill people on #11 green, so they took a detour around to the right to create #6 and #7 as distinct fairways ... because there was a great hazard keeping them from extending #11 green to the left to use as #7 and avoid a crossover!