What is the canonical coordinate system of a roll of tape?

Unroll the tape on your right index finger, starting from where your thumb meets your index. That's the origin. Your thumb points along the positive Z direction. Your index points along y, and x is your middle finger points along x.

Rolled or unrolled?

A small wad, after getting stuck to itself and an unsuccessful attempt at separating it, that is now stuck to one of your fingers no matter how hard you attempt to flick it off.

You have to know these things when you are a king.

Fascinating! Seems like that could be an effective way for hobbyists to attach BGA components to a PCB without a reflow oven - any idea if that’s been tried?

It has been tried and doesn't work very well, even for low-density pinouts:

https://tomverbeure.github.io/2019/11/21/Z-tape.html

From the datasheet linked there, it looks like the minimum recommended spacing to ensure electrical isolation between adjacent conductors is 15 mil (0.4mm). In an example I found here [1], the pitch between BGA pins is 0.5 mm to 1.0 mm, with pad sizes roughly half of the pitch. So it looks like for the larger pitches, it might just work. 3M also says they'll do custom tape with better isolation, but that might just be more expensive than a reflow oven.

[1] https://www.nxp.com/docs/en/application-note/AN10778.pdf