In Hempstead, and in other high-tax, high cost-of-living communities across the country, tax bills routinely run far above the new $10,000 threshold once state income and local property taxes are taken into account. In Nassau County, which includes Hempstead, the average state and local tax deduction in the county, including property taxes, topped $20,000 in 2015, among the highest in the country, according to data from the I.R.S.

The new cap does not take effect until January, however. The tax bill explicitly prevented people from prepaying state income taxes, but it did not address prepayment of property taxes. That gave homeowners a brief window to pay their 2018 property taxes in 2017, and to take the full deduction when they file their federal returns this spring.

Officials in Chicago, Washington, Fairfax, Va., and other communities reported huge surges of residents prepaying taxes, often showing up in person, checks in hand. Democratic politicians, who have opposed the bill, egged them on, arguing that the bill targeted states that tend to vote for Democrats.

Even before President Trump signed the bill into law last week, local officials in Washington announced they would accept prepayments in what the city’s mayor, Muriel Bowser, called a bid to “protect Washingtonians from the negative impacts of this devastating legislation.” Gov. Andrew Cuomo of New York last week signed an executive order opening the door for prepayment, a move he freely described as a bid to circumvent the new law. Local officials in Maryland, Virginia and other states made similar moves.

Even Chris Christie, the Republican governor of New Jersey and an early supporter of Mr. Trump’s, on Wednesday signed an executive order instructing local officials to accept prepayments.

The I.R.S. memo, however, threw many of those efforts into question. The memo said that property taxes paid this year would be subject to the old 2017 rules — but only if the taxes are actually assessed in 2017. That means that payments based on estimated assessments, or for years further in the future, probably would not qualify for the deduction.

Final answers might not be available anytime soon. The I.R.S. guidance left plenty of room for interpretation and was, in any case, nonbinding.