Okay
Right hand should work in all cases.
Right hand, palm up, thumb to right, index away from you, middle finger straight up.
Thumb is X positive
Index is Y positive
Middle is Z positive
This applies to pen and paper, and milling machines
Move so that index finger is up. This applies to CAD/CAM systems, where Z plus is coming at you from the screen (default view, mostly.) X (thumb) is still pointed right.
Turn (contort) so that the index (still Y) is pointed down, middle is pointing right and thumb is towards you. This applies to lathes with the turret working from the operators side of the spindle.
This is why on these machines, G02 and G03 as well as TNRC seem backwards.
Contort some more. Index points up, middle points right and thumb points away from you. This applies to lathes that have the turret approaching the work from the side opposite the operator.
Here, G02 and G03 and TNRC seem correct.
Haven't had experience on HMC's so I'm not familiar with that configuration, but I'm sure it agrees with convention.