Let me first say I'm not a surveyor or an expert on the use of Parcel Fabrics.
A basis of bearing is not a rotation factor for a parcel fabric. It applies individually to each plan/plat/whatever as a source going into the fabric. Old 10.0 help file on the specific tool describes some of it, and portions of that got incorporated into this newer 10.2 help page. Note that it points out it's not a grid to ground correction.
It appears you know what a basis of bearing actually is in broad terms - a stated line with a stated bearing. When comparing to any other data set, you can look at the same line's bearing in that, and the difference between the two is theoretically your correction factor. However I almost never find this to be all that useful, as even correcting to match up that line, other lines that can be cross-referenced will be off by a different amount (usually both length and bearing, and we haven't even brought up scale factor correction yet). This is probably a pretty good indication of how little I actually know about the proper use.
Coming back to Parcel Fabrics. Essentially a parcel fabric will warp any input description/boundary to fit established control points in the fabric. I have not used them extensively but I know there are rankings to points - some are held as absolute when running adjustments, and some can be moved and modified with those adjustments. There is a basis of bearing section at the bottom of the About the parcel fabric adjustment page that might give you some more detailed info.
Once something is in the fabric though, it's governed by the fabric and its control points. A fabric adjustment is less about moving a single subdivision and more about creating a best-fit between all of the conflicting descriptions of parcels within the fabric. You enter a parcel's actual description, and then fit it to the fabric by saying what corner lines up with something already in the fabric. It stores the input description values and uses that new information to remap/warp the existing info to provide best fit (so basically every parcel has two descriptions - the inputs, and what it got warped to when fitted). You might check out the tutorial if you haven't.
Someone else more expert in fabrics may be able to provide a better answer (or shoot holes in mine) but let me know if I can attempt to provide any more clarification or information. Hopefully it will at least move you in the right direction.
