Your goal is "pivot the blade to make sure it closes and everything lines up." This is two separate steps:
(1) Rotate the blade (curves, lines, solid, whatever your CAD model is built from) about the pivot point.
(2) Detect any component interference or misalignment.
#1 should be simple enough. All CAD programs I've played with have a rotate tool; you simply select your geometry, select the center point, and proceed in one of several ways.
#2 is not so simple - are you expecting the CAD program to do this for you, or will you simply look at your design to check for interference? If you want the CAD program to do it, you'll need a 3D program and you'll have to "mate" the components together (i.e. "specify relationships" as mentioned above). Conversely, if you can do it by eye, all you need is a simple 2D CAD program. I use SpaceClaim for 3D and AutoCAD light for 2D.