Nevertheless, had she not initially felt enough respect to want to understand what was important to her husband, she wouldn't have made the shift.
Could someone explain the structure used in this sentence and put it another way?
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That sentence expresses, by means of inversion, the same as this counterfactual or past hypothetical conditional:
The point is that she did have that feeling and, as a result, she made the shift.
There are three main types of inversion with conditional sentences:
If you should have any doubts, please do not hesitate to contact us -> Should you have any doubts, please do not hesitate to contact us.
If I were to discover that he is guilty, I'd send him to prison -> Were I to discover that he is guilty, I'd send him to prison.
If I had known you were in trouble, I would have helped you -> Had I known you were in in trouble, I would have helped you. (This is similar to the case you presented.)