"Should" as formal IF

Can a phrase starting with "if" always be replaced with a phrase with a phrase starting with SHOULD? It doesn't seem like we can make a direct substitution. For example, we say

If he askED you to move to Italy, would you do it?

But

Should he ask you to move to Italy, would you do it?

Why is this, and what is the rule for the substitution?

The first option is correct, the second not. The verb form, should ask is subjunctive and expresses a hypothetical situation that has not yet happened.

The verb form should asked is not a standard form. If you were describing a hypothetical referring to a time already passed, you could say

Should he have asked you ...

or

If he had asked you ...

But note that these describe something that did not happen.

Modal verbs require verb complements in the infinitive form:

Should he ask ...
He should ask ...

Should SUBJ asked is ungrammatical.

Asked could occur, however, if the complement of the modal is auxiliary BE or HAVE taking asked as its complement:

Should he be asked, he will respond ...
Should he have asked, he would have been told ...

The last one, however, is clumsy; with an irrealis we'd usually say

Had he asked, he would have been told ...