It's probably best to think of it as a priority system rather than a set in stone rotation. I don't main NIN so I'm sure I'll be corrected if wrong - but:

Get Huton up before pull, and then your priority goes:
Armor Crush* > Dancing Edge > Shadow Fang > Mutilate > Aeolian Edge

Armor Crush takes some getting used to, but generally you want to refresh Huton with it somewhere between 20-40s left on it. You have some leeway here as ideally you wanna make sure your other 3 dots are up before refreshing, but downtime mixes things up.

Dancing Edge combo takes about 8s, so start aiming to refresh around that time. Also if it gets to Aeolian Edge but you can't hit the rear then Armor Crush is stronger (flank only), Dancing Edge is also stronger (front and flank, but Crush is better for flank). Always use Crush when you need to regardless of positionals. Always use Duality with Aeolian if possible, but if it seems like you won't be able to get one off any time soon just boost Armor Crush. Obviously WARs might be using Storm's Eye in which case you don't need to worry about Dancing Edge, that said it's still stronger if you can only hit the front.

For Ninjutsu, unless you need to save Trick Attack for an upcoming burst phase or something, use Suiton -> Trick Attack when available, otherwise Raiton for single target (if your mudras are a bit laggy you can use Shuriken instead). That's a very general run-down at least.