Aligning multiple data sets to a standard direction, with step changes

I have a large number of data sets, that have time in the x column and direction in the y column. I would like to have all of my data align to a reference direction, so that they all match each other into 15 degree segments.

What I want to do to start is look at all the sets of data that I have, and see what the difference between the directions is at a common point. Then, take, the most common direction as the correct one, and correct the off data points to that reference direction. Some of the data that I have I know is off by ~180 degrees for example, and I will need to correct it to the right direction.

Is there anyway this can be done to check for a step change in the data sets? We wrote a code that checks the direction of data set 1 to 2, data set 2 to 3, and data set 1 to 3 for a sample with 3 data sets. It corrects the data set that is off to the other two data sets that are closer in direction (both are within 15 degrees of each other). However, this only works at the beginning but gets messy towards the bottom of our corrected set. Also, we don't know how to expand this to include a large number of data sets without rewriting a lot of code.