Carolyn Kaster, Associated Press
President Barack Obama, joined by House Speaker John Boehner of Ohio, left, and Senate Majority Leader Mitch McConnell of Ky., right, speaks to media, Tuesday, Jan. 13, 2015, in the Cabinet Room of the White House in Washington. Several misperceptions have cropped up in the heated debate over Congress' struggle to provide money for the Department of Homeland Security and avert a partial shutdown at week's end.

WASHINGTON — Several misperceptions have cropped up in the heated debate over Congress' struggle to provide money for the Department of Homeland Security and avert a partial shutdown at week's end.

Many Republicans want to halt the funding unless President Barack Obama's liberalized deportation policy is rescinded. Senate Democrats blocked that effort, and House Republican leaders signaled Tuesday that they might rely heavily on Democrats to end the impasse.

A look at some of the statements:

1. Assertion: House Speaker John Boehner and Senate Majority Leader Mitch McConnell — Congress' top Republican leaders — took separate uncoordinated tracks on Homeland funding.

In fact, both men employed the same strategy. They endured public derision by repeatedly pushing bills that had no chance of being enacted. The purpose was to show hard-core conservatives they're unable to force Obama and congressional Democrats to back down on immigration.

McConnell held four roll-call votes, over four weeks in February, all blocked by Democratic filibusters. Democrats accused McConnell of wasting time, accurately saying the fourth vote had no better prospects than the first.

"We all know this is going to end with a bill funding Homeland Security" with no strings attached, Senate Democratic leader Harry Reid said on Feb. 3, after the first vote. "The Republicans should stop the posturing for the right wing of their party."

Boehner uses the same painstaking "prove the futility" strategy to deal with conservatives who demand that Obama bend to their wishes on taxes, immigration and other issues. Sometimes at the last minute Boehner passes contentious bills largely with Democratic votes. That lets his hard-core conservatives vote "no" and stay popular back home.

That happened on the "Fiscal Cliff" tax showdown and last year's debt ceiling increase. But in 2013, Boehner let House conservatives trigger a temporary government shutdown. Polls said it hurt the GOP's image.

On Tuesday, Boehner said he will let the House vote on the Obama-backed, no-strings-attached bill to fund Homeland Security, which presumably could pass with numerous GOP defections but Democratic support.

2. Assertion: The Republican Party is showing it can't govern despite holding House and Senate majorities.

Talk of "the Republican Party" in Congress — and especially the House — is sometimes unhelpful. The House GOP caucus is badly splintered, with a sizeable number of ideological conservatives willing to defy the party's leaders time and again. Last Friday, 52 House Republicans humiliated Boehner by helping kill his plan to fund Homeland Security for another three weeks.

These conservatives tend to focus intensely on dogmatic conservative voters back home, who dominate their primary elections. They worry less about GOP efforts to win presidential elections, or national polls gauging Americans' attitudes about the party.

GOP Rep. Devin Nunes of California said party leaders expend important energies trying to "placate a small group of phony conservative members who have no credible policy proposals and no political strategy to stop Obama's lawlessness."

In a vivid sign of the party's divisions, a political group allied with House GOP leaders says it will spend more than $400,000 this week in advertising aimed at pressuring many of the 52 House Republicans who defied the leadership Friday.

After Mitt Romney's 2012 loss to Obama, the national Republican Party commissioned a study of what went wrong. Its only policy recommendation called for the party to "embrace and champion comprehensive immigration reform," to appeal to Hispanic voters.

Most congressional Republicans ignored it, and they've strongly opposed Obama's immigration policies to spare millions from deportation.

3. Assertion: Republicans control the Senate, by holding 54 of its 100 seats.

Any Senate majority short of 60 seats is not control. Both parties rely heavily on filibusters, a delaying tactic that can block legislation indefinitely with 41 votes.

It's possible that Republican leaders would be facing less criticism on the Homeland Security issue if they had been more vocal about this reality after winning the Senate majority last fall.

4. Assertion: Those November elections determined Congress' makeup.

Technically, yes. But hundreds of House elections are decided in the party primaries, which take place throughout the election year, often with modest media and public attention.

So many House districts are overwhelmingly liberal or conservative that a Democratic or Republican nominee, respectively, has no realistic chance of losing in the November general election. The primary is everything.

Primary voters tend to be more ideological than general election voters. A big problem for Boehner, on Homeland Security funding, is that many House Republicans answer to voters who strongly dislike Obama, illegal immigration and federal bureaucracies. For them, a partial DHS shutdown designed to pressure Obama on immigration is a political winner.

5. Assertion: House Republican rebels who defy their leaders generally have little seniority and tenuous ties to the party's power structures.

That's partly true. But the 52 who voted against Boehner's effort Friday to fund Homeland Security included some influential veterans. One was Joe Barton of Texas, a 30-year House member who once chaired the Energy and Commerce Committee.

Another was Lamar Smith of Texas, a 28-year House veteran who has chaired the Judiciary and Ethics committees.