All you have to do is use a CNC machine once to get your X,Y,Z's straight as the consequences of getting them confused on a stupid machine is often expensive and catastrophic.
As a quick point of reference Z is always the direction spindle is facing. X is horizontal and perpendicular and Y is 90 off X. A,B,C are rotation movements and such off each plane.
So a lathe is Z on the spindle and X on the crosslide. But to confuse that if you get live tooling it usually means the Z (spindle) can index or be controlled other than just rpm turning it into C axis, this helps with your live tooling which moves in Y to do mill turning.
Having fun yet? If you ever wonder why Bridgeport mills have been around so long just count the axes:
X...table left right
Y...saddle in out
Z...knee up down
?...spindle up down
?...Turret rotation on column 360 degrees
?...ram in out
?...head head nod 45+ forward back
?...head rotate 180+ left right
You start to add all those up and you get a machine that can get into a lot of funky spots. Makes them useful for when you get a guy who insists you cut a classic style breather tube for the intake manifold of his streetrod, right away while you are in the middle of a move and your machines are packed away. Yes this cost him more than the manifold was worth.
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Luckily it turned out fine. Gave a new meaning to cramped work space. I used every axes but the turret.
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