The spinners action is based on which way the ball would spin if the bowler was bowling to a right handed batsman. Left-arm chinaman is equivalent to right-arm offspin.
No it isn't.
Left-arm chinaman is the equivalent of leg spin, whilst Slow left-arm orthodox is the equivalent of off spin.
Just developing this, Orthodox bowling for right handers is changed to un-orthodox for left handers and vice versa.