At the meeting of the Transcontinental Railroad in Utah on May 10, 1869, there were two locomotives, one from the east and one from the west. Obviously, one of them had to back up. Which one backed up?
Probably the one that didn’t have as far to go to a siding.
Ambrose Bierce has a book on this. My recollection is that the West won the race.
The National Park Service has some interesting information at its website for the Golden Spike National Historic Site on this page it says
So it looks like they both backed up. Which isn’t all that surprising, when you consider the intense rivalry between the UP and the CP. I don’t think either firm would care to cede any honor to the other.
It looks like passengers changed trains at Promontory until December of 1869 when Ogden became the official terminus where passengers had to change rail lines.
In an issue of Mad in the early Sixties, this ceremony was spoofed in a six-panel cartoon by George Woodbridge. It shows the locomotives approaching; California governor Leland Stanford pounding the spike into place–and two of the four rails touching instead of four! 
Paging Dr. Freud!
Bet you mean Nothing Like It In the World : The Men Who Built the Transcontinental Railroad 1863-1869, by Stephen E. Ambrose.