Which sentence is grammatically correct?
1. I had better called my mom back.
2. I had better call my mom back.

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@Carly Good link, I never even heard about modal verbs until I got on italki 😅. How did you learn about all these?

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The 2nd one. With have /has/had (perfect tenses) we use, generally, 3rd form of verb in a sentence unless you're using it as a verb. But here inthis case HAD BETTER IS A MODAL VERB SO WE USE THE 1ST FORM. If you're using it as an adverb or noun or an adjective, then the case would become different.

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#2 & you can say “I’d better” or “I better” to make it more natural sounding

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Yup, "had better" is another world of "should" and it is used more to express urgency or threat. 
https://www.youtube.com/watch?v=9r88YWp9XQo

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The verb doesn't follow the had + past participle rule because it's not acting as a regular verb. It's called a modal verb. 
https://learningenglish.voanews.com/a/everyday-grammar-you-had-better-learn-modals/2865365.html