I had better leave right away.

I had better leave right away.

What part of speech is the words “had better”?

If HAD BETTER = SHOULD then can we call it a modal verb?

As I understand it, ‘had better’ is an expression meaning ‘very/highly desirable/necessary’. Yes, it has the force of the modal ‘should/must’, but not a modal.

Thank you, Anglophile.
If you had to analyse that sentence how would you parse “had better”?

It is really an interesting question. I could only say that ‘had’ is a verb and ‘better’ an adverb; both together form the expression ‘had better’.

Honestly speaking, our great James would be able to do more justice to your question (parsing). Let’s hope he will come to our rescue.