Perhaps I'm just overlooking the explanation in grammar references (and questions on ELU), but I haven't found a discussion of this construction (or its advisability): a verb is omitted after a modal, but the verb form omitted differs from the verb form after the conjunction.
Standard:
Cutting for sectioning can be and has been done very effectively by conventional gas and oxy-arc techniques. Coast Guard Engineer's Digest, Issues 205-211, p.20 (1980)
On a more specific level, relevance can be and has been considered in relation to specific types of knowledge communications. K. Sparck Jones and P. Willett; Readings in Information Retrieval (1997)
or "You can and should say something."
Versus:
It is here that H. Richard Niebuhr argues that religion can act not only as a function of dominant culture but that religion can and has developed its own initiative that can move us toward good sense. Melanie Johnson-DeBaufre et al.; Common Goods: Economy, Ecology, and Political Theology (2015)
Much can and has been said about Lincoln's word choice ..., such as his decision to use the noun expiration in relation to a quantum of time in the context of war, but what captures my attention is... Beate Hampe; From Perception to Meaning (2008)
Dear Kenneth,
... I'm pleased, nach (natch), that you like the mutifenestrated house. You pick out the pieces that interest me most (though as I should have but didn't say in a footnote to the Whitman piece, the attention to his ... Paul Jay (ed.); The Selected Correspondence of Kenneth Burke and Malcom Cowley, 1915–1981 (1988)
"... When Larry (a recently discharged patient on her unit, with whom she had been quite friendly) left last week, I didn't feel anything either. I should have, but didn't and he had nowheres to go." Michael Stone; The Borderlines Syndromes (1980)
Early examples:
The most eminent and sober Nonconformists, as it can and has been proved, have own'd Her to be a true Church; ... John Harrison; The Minister of Cirencester's Address to the Dissenters of his Parish (1698)
And I must take the Freedom to tell my Author, that it both can, and has been shown before ever he was born, That there was more than an Occasional Submission, by the Congregation at Antioch, unto the Apostles and Elders of the Congregation in Jerusalem. Thomas Aiton; The Original Constitution of the Christian Church (1730)
Is there a term for this construction or general phenomenon?