Learn German: Ich hätte meine Frisur nicht ändern sollen. - I should not have changed my hairstyle.

This is the double infinitive rule. When you use a modal verb like sollen in the past perfect tense along with another verb like ändern, you don't use the past participle. Instead, both verbs stay in their infinitive (dictionary) form at the end of the sentence.

Why do we use hätte here instead of habe?

We use hätte (the Konjunktiv II form of haben) because this expresses a past regret. You actually did change your hair, but you wish you hadn't. Using hätte combined with a modal verb like sollen translates to what you 'should have' done.

Is the order of the two infinitives at the end strict?

Yes! The main action verb (ändern) comes first, followed by the modal verb (sollen) at the very end. The modal verb is the core grammatical piece of that cluster, so it takes the absolute last spot in the sentence.

Why does nicht come right before ändern sollen?

In German, nicht usually goes right before the specific part of the sentence it is negating. Here, it negates the entire action of changing your hair (ändern sollen). When negating verbal phrases at the end of a main clause, nicht sits directly in front of them.