First question is WHY would you take apart ALL the splices???

Very interesting.

Switch B receives the power from A on the travelers and returns it on the common.

Switch A receives power from Box A (switch loop) on the common and sends it to switch B on the travelers which sends it back to to switch A but is not connected to said switch, it gets connected to the other wire in the 2 wire cable which sends it back to Box A

The white black and red in Boxes A and B could be feed through to the switches.

With a question put like that, I wouldn't even think of trying.
With a bit of logic and a understanding of basic wiring practices, and maybe a continuity tester, you can probably figure it out yourself.

I agree Randy. Only I say just call an electrician. It's worth the service call fee for the peace of mind that it is put back together correctly and safely.
Sometimes we have to pay for and learn from our mistakes.

The logic behind ripping something apart and not paying attention to what's what eludes me.

Just be aware, when you hire a professional electrician, you are paying him for his knowledge, truck full of tools and testing equipment, insurance, license, years of education and field experience. Please don't pull the "Why did you charge me so much to fix something that is so easy" when he is finished. What may take a homeowner countless hours or even days can take a qualified electrician mere minutes or just a couple hours to diagnose and repair. That is what you are paying for.