Does a bicycle axles direction always have the writing stamped on it facing forward? My bottom bracket adjustable cup keeps loosening even though the lock ring is tight. I’m thinking maybe I reversed the axle in the wrong direction

It shouldn't matter.

For hub axles, there may be more threads on the right side of the rear.

For bottom brackets, the right side of the spindle may stick out more (longer from bearings to right).

Do you know what "type" of bottom bracket you have?

French? English? Italian? Swiss? ISO?

French and Italian have right hand threads on both sides. I think this normally has the "fixed" cup on the right (drive side).

I've used Purple Locktite during assembly in the past.

Shimano 600 cup and cone

+1. Imagine sitting on the saddle, looking down at the bottom bracket. The engraving will read left to right.

I imagine sitting on the saddle and a seatpost and seat tube and bottom bracket shell being right below me.

I'd carefully clean the offending cup and try the Loctite Purple on it.

A reversed axle won't cause the adjustable cup to loosen.

I assume you're tightening the lockring on the adjustable cup sufficiently?

My first guess is that the lock ring is not as tight as you think.
Second guess is a large burr on the lock ring that bottoms out before the lock ring seats properly on the frame.
Third and fourth are partially stripped threads somewhere that barely hold and an extremely non square BB shell.

I do not advise using thread locker on bb threads. It shouldn’t be required and can be a real bugger to reverse.