Huh. I guess at this point I don't remember exactly what I coded into 'rebuild as global sequence', I just thought it wasn't right.
The righty-to-lefty is simple once you get it figured out, but fairly unclear in terms of what to do. It is embedded inside Tools -> Mirror - Mirror X. Normally this feature only reverses selected meshes/geosets (which would totally ruin their links to the bones and stuff), but if you go to the View Controller window that opens when the program starts, click the purple box at the top of it, and toggle on the Show pivot points option, it lets you select and move bones along with the mesh.
When you mirror having both the bones and mesh selected, it will not only flip both of their positions across the mirror but also flip their animations. It's really nifty if you figure it out, but it really should be documented more clearly.