CHERRY HILL, N.J. — New rules unveiled by the IRS undermine some states' attempts to help residents keep a version of a popular federal income tax deduction. Last year's Republican tax overhaul put a $10,000 cap on deductions for state and local income taxes, which could affect some high-earners in high-tax states.

A handful of high-tax states sued the administration earlier this summer over the Republican tax law, which they say was in part politically motivated to attack Democratic-leaning states. Republicans have said those states should reduce their taxes instead of fighting the administration.

Here are questions and answers about the IRS rule announced Thursday, what it means for states and how high-tax states might respond.

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WHY IS THE DEDUCTION SO IMPORTANT IN THESE STATES?

The SALT deduction is popular and widely used in high-tax states, including California, New Jersey and New York.

In those places, many residents have state and local tax burdens of more than $10,000. In some cases, much more. A cap on the deduction means they will not see the same tax break as people with similar incomes in other states. Some will see tax increases because of the cap.

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WHAT ARE STATES DOING ABOUT IT?

About a dozen states have adopted or considered laws this year to help residents get around the cap. The most common way is by allowing donations to government entities in exchange for state or local tax credits. The idea was that people could still deduct their charitable contributions from their income for federal tax purposes.

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WHAT DOES THE IRS RULE MEAN FOR THOSE PROGRAMS?

It thwarts them. The rule allows federal deductions only for the portion of donations not subject to state or local tax credits.

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DO THE STATES HAVE ANY RECOURSE?

That's not clear. Connecticut, Maryland, New Jersey and New York sued the federal government last month over the cap, alleging it was aimed at hurting some largely Democratic states and trampled their budget-making authority.

New York Gov. Andrew Cuomo and New Jersey Attorney General Gurbir Grewal said separately on Thursday that further legal action is possible. In a statement, Cuomo said eliminating full state and local tax deductibility will cost New York families $14.3 billion a year and that the effects already are being felt, citing declining home sales in some wealthy communities.

Grewal said the new IRS rule will "make it more difficult for states like New Jersey to cope with the backward tax policies the federal government imposed on us last year."

Officials in Connecticut and Maryland said they were reviewing their options in light of the new rule.

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WHY DON'T THOSE STATES SIMPLY CUT TAXES?

That's a question some Republicans are asking, saying it's the states' problem if their high taxes lead some wealthier residents to lose out under the Republican tax law.

The high-tax states generally offer more public services. And some of them are in perpetual budget squeezes — for instance, needing to catch up on payments to their pension systems for government workers.

They also contend that having the federal government tax earnings that are paid to state or local governments constitutes double taxation, and say they are challenging the new deduction limits as a matter of fairness.