So I thought that same alignment restriction would always apply on a given item with a given enchantment.
But... I got two Silver Helms + 6. One usable only by good, the other only by evil...

Apparently alignment restriction is something that can be completely randomly applied to a particular item.

Yeah it's kind of annoying like that. You can use the Enchant Item spell to not only improve an item but also randomly change its alignment restriction, if you want. Alternatively, you can change your character's alignment (the easiest way is to slip towards evil by fighting lots of monsters without giving them a chance to accept a bribe), equip an aligned item, and then cast restore alignment to bring the character's alignment back. As long as you don't un-equip the item, you can keep using it.

if you need to bring your alignment up (i.e. make an evil-aligned character become good) you will have to offer monsters bribes a lot. This can get annoying because sometimes they'll want to take all of your gold or something.

Yes, I found playing around with alignments too annoying so I just leave them as it is... made me miss out some cool items, but I just tell myself I'll always find better.

I didn't bother changing my characters' alignment on purpose much either, but they would often accidentally slip to evil after doing lots of fighting, so I was sometimes able to give them items they couldn't use before.

Without spoiling too much...

Any equipment that you find at or above a certain enchantment level (I think +5) has an alignment, so as you fight better monsters and get better treasure, you'll run across this problem more often. But there are extremely easy ways to set your character's alignment at the drop of a hat, and as you explore and experiment, you'll eventually find how to do it. So if you find truly exceptional items that your character can't wear because it's misaligned, it could be worthwhile to store it with an unused hireling rather than selling it.