If they’re soldered together, then they’re connected. Just treat them as a single ‘common’ and you’re good.
This part’s easy. Pick a button, and look to see which of the two wires going to the button is connected to the common line with the rest. Mark it with a sharpie or a piece of electrical tape. Plug it in, preferably into the xbox. With your multimeter set to detect Volts, and whatever setting is just over 5 volts (on my multimeter, that’s the 20 volt setting) touch the black probe to the wire you marked with a sharpie, and the red probe to the other wire going to that button. Touching the QD’s on the button is fine, but the button can not be pressed. Look at your multimeter. The reading should be either around 5 volts, or around -5 volts.
If the reading is 5 volts-ish, it uses a common ground, and your inability to wire the p360 just means you pulled a stupid somewhere and you should check your wiring.
If the reading is -5 volts-ish, the uses a common high setup. You can still use a p360 just fine, but you’ll need to invert the four direction lines. The p360 uses low for pressed, and high for not pressed, the opposite of what a common high uses. It’s a simple thing to do, and takes just one chip, like a 74HC04 inverter. Let me know if you need help figuring out how to wire the chip, but you’ll probably figure it out.
If you shipped it, I know I could figure it out, but there’s no reason you can’t figure it out with the test above.
Picture ain’t gonna help.