Is there a way to realign a Basemap to known reference points?

I am pretty new to ArcGis. My company got it, too assist with our survey department. We have version 10.2.2.

I have figured out how to bring in Basemaps and assign them to the proper state plane coordinate system and then bring in AutoCad drawings with the same State plane coordinates system assigned too it.

The problem i am having, is we get drawings from time to time, that have been adjusted from grid to ground using a 0,0 base point of adjustment. this throws the Cad drawing off from overlaying the Base map by 100 feet or so. I would rather not readjust my cad drawings, but instead move the basemap to overlay the cad drawing.

Most sites we work on are pretty localized. Involving no more than 1000 feet across. In the Cad we generally have buildings mapped up or other structures that i can see in the aerial Basemap i have brought in.

So i was hoping i could realign the basemap using two similar adjustment points in the basemap to match up with the Cad drawing.