As House Republicans continue to press Democrats this week for even bigger cuts in a bill to avert a government shutdown, it is likely that more and more members of Congress will face constituents who, while supportive of the concept of cutting federal spending, do not care much for the specifics.

Such inconsistencies, while hardly new to this Congress, are political chum for Democrats.

“You cannot vote to cut veterans’ benefits in Washington and then go pose for pictures with veterans back in the district,” said Representative Steve Israel, who runs the Democratic Congressional Campaign Committee. “There is a pattern of duplicity here, and we’re going to make sure it comes back to haunt them.”

“Pattern” may be an overstatement, but there are several examples of members’ voting for cuts that they then deplore. Representative Bobby Schilling voted against rail financing for his district in Illinois, and later said that he did so only because he knew that the Senate would not sign off on the cut.

The Republican spending bill made large cuts to the Department of Energy, "regardless of mission,” and including the National Nuclear Security Administration. Several Republicans, including Representative Michael R. Turner of Ohio, wrote to Representative Paul D. Ryan, Republican of Wisconsin and chairman of the House Budget Committee, seeking to restore money for the agency.

“Congressman Turner voted in favor of H.R. 1 as it was the beginning of a longer conversation on how our nation is to proceed with its finances,” said his spokesman, Thomas Crosson. “In the time since, Congressman Turner has spoken with Chairman Ryan and leadership to stress the importance of N.N.S.A. funding, and its implications for our national security.”

Also potentially harmed by cuts were research centers in Illinois, which caught the attention of Representative Randy Hultgren, a Republican from that state. Mr. Hultgren visited Fermilab, which conducts research in high-energy physics, and pledged support, according to a spokesman for the lab and news media accounts. Andrew Flach, a spokesman for Mr. Hultgren, did not return numerous calls.

The Republican budget bill also called for cutting a type of popular transportation grant. Since the cut was proposed, there has been an apparent rush to get work going on projects financed by the grants to keep them from being ended by any budget-cutting deal agreed to by the two parties and the White House. Consider the case of the Memorial Bridge, which stretches over the Piscataqua River between Portsmouth, N.H., and Kittery, Me. The aging bridge, in ill repair, was set to receive $20 million in transportation grants.