Looks like they were trying to install in level or close. The had to be pretty stupid to try that with that long a run and yard slope. If the yard drops 3' you will have a 3' fence at the yards high point to have 6' at the low point, who would want that.
I'm guessing they realized they were WAY off on bidding it and just bailed the job. That and realized the had to backtrack to save it. With the poor workmanship why would you give them another penny.
On the plus side that looks like a fair amount of lumber and if you can live with the posts being set where and how the are, or maybe at least fix, you probably got a fair amount of the moneys worth. Of coarse the aggrevation has a cost. But I bet if you figured out whats there, its probably at least 2k or even more, hard to tell from the picture how much there is.
The fence I just finished was half the length and build abit different with a more complex design and cedar, but lumber went near $4500.
It will be hard to fix the screw up though and look at it every day and learn to like it.
I would if you are willing to take the time.... take her apart. Carefull beat the pickets off, de-nail. Beat the cross members off, de-nail. At worse you have extra nail holes, use that lumber away from the house....you'll probably never see it. Use the new stuff closer to the house, by then you'll have a rythem and be a pro.
You could even dig the posts out and cut the concrete off the bottom. If they didn't use much and it looks like there is better than a foot on top. Plus with that style of fence you can stop the post at the top cross member. So about 5.5' out of the ground and 3' in. If they used 10' posts you may be able too, ALOT of work though. Judging by the pictures they would have to be 10' posts, that or they are only 1' in the ground.