Here's my approach to this issue. :-
I have a tap-holder (jacobs chuck) and two different sized die-holders on 1/2" shanks, and A pair of MT adapters for my lathe tailstock and mill spindle – on the drill-press I just use the chuck jaws. This set-up will deal comfortably with thread-sizes ranging from 12 BA to 3/8", and can be used to start larger threads which can then be finished by hand. If a non-cylindrical thread's position and alignment are critical, I'll usually set it up in the milling vice to ensure accuracy.
Use in lathe tailstock (the die holder is hollow) :-
Mill spindle :-
Drill-press :-
I too have suffered from the dread 'image rotation' issue, my apologies.
Edited By JasonB on 30/03/2022 14:50:40