President Obama faces a pair of critical tests Tuesday, and how they turn out will tell us something—not “everything” but something important—about the political hit he is taking as a result of the jobs crisis.

They will also offer indications about whether Democrats have really figured out how to use their support of Social Security, Medicare and Medicaid—the most powerful weapon in their political arsenal—to win elections.

The pair of special elections for open seats in the US House, one for a historically Democratic New York seat in New York City and the other for a Republican-leaning seat in Nevada, will send signals about confidence in the president’s stepped-up focus on job creation versus the Republican austerity agenda.

No special election is ever a perfect indicator of the national political mood. But these two offer legitimate measures.

If Democrats win both seats—with messages that attack Republican threats to the social safety net, especially for the elderly—then these two special elections will serve as extensions of the March special-election win by Kathy Hochul in a western New York district that had been held by Republicans for decades.

If Republicans win both seats, with “blame Obama” messaging, then the already considerable pressure on the president and his partisans could increase considerably. And there will be some evidence that Democrats were damaged when they muddied their message (by suggesting that cuts in Social Security, Medicare and Medicaid might be “on the table”) during the debt-ceiling debate—making it harder for them to use what looked to be their best message as a 2012 electoral cudgel. If that’s the case, they are in serious political trouble.

Here are the particulars:

NEW YORK 9th: Sixth months ago, Congressman Anthony Weiner, the scrappy liberal Democratic representative from this Brooklyn-Queens district, was emerging as a national spokesman for his party and his ideology. Now, he’s a political punch line and his district could fall to a Republican. Polling suggests that GOP nominee Bob Turner, a former media executive, running even with (perhaps even a bit ahead of) Democrat David Weprin, an old-school machine pol whose nomination was engineered by party insiders who did not want to run a heavy-hitter for a seat that is likely to be threatened by redistricting. The assumption Democrats could nominate “anyone” and hold Weiner’s seat was grounded in political history: Former House Judiciary Committee chairman Emanuel Cellar took the seat for the Democrats in 1922 and it has been held since by some of the party’s biggest names: Cellar, Elizabeth Holtzman, Chuck Schumer and Weiner. Had Democrats simply nominated Holtzman, who remains an active and engaged player with considerable appeal, they could have easily held the seat.

But the party went “small” and is threatened. Much is being made of this race as a test of Jewish sentiment regarding the Obama administration’s Israel-Palestine policies. There’s something to the argument, and former New York Mayor Ed Koch and the conservative New York Post have tried with all their might to exploit the issue. But Turner’s not Jewish, while Weprin is an observant Orthodox Jew who had put considerable distance between himself and Obama regarding the Middle East. Normally, that would be more than enough to dial down any concerns. But Weprin’s weakness as a candidate and Obama’s weakness on the ground even in New York (one poll suggests his approval rating in the district is just 42 percent) have made a mess of things.

To counter a GOP surge, Democrats have dialed the volume way up on the Social Security, Medicare and Medicaid issues. If Weprin wins, the message will be a simple one: even when things go awry, focusing on the defense of the social safety net remains a winning message for Democrats. If Weprin loses, the message will be more nuanced but equally important: first, Obama has not turned the corner on the jobs issue and made it his own, even with traditionally Democratic voters. He will need to step things up dramatically and Democrats will need to get even more aggressive on the Social Security, Medicare, Medicaid front. They were slow to put money into messaging on these issues (unlike in the Hochul race); they need to be a central focus all the way through every critical contest. Second, at a point when Obama and the Democrats are vulnerable to the blame game, local bosses cannot nominate placeholder candidate to carry the party banner in elections that are likely to be read as signals regarding the strength of the national party and its president.

NEVADA 2nd: After Senator John Ensign, R-Nevada, was taken down by his own sex scandal, the state’s Republican governor appointed GOP Congressman Dean Heller as a replacement. Under the absurd reading of the Constitution favored by governors around the country, open US Senate seats are filled by appointment, while open House seats must be filled by special elections. There should have been a special election for Ensign’s Senate seat, but that did not happen. There will be a special for Heller’s seat on Tuesday.

Since this seat was created in 1982, it has been held by Republicans. But Nevada has been hit especially hard by unemployment and the as-yet-unresolved mortgage crisis. That should have made this a ripe race for a candidate with a Hochul-style message and focus. Democratic nominee, Nevada state Treasurer Kate Marshall, has made a reasonable effort, but she’s not been as aggressive or focused on establishing the distinction between a pro–Social Security, Medicare, Medicaid Democrats and a cuts-inclined Republican. In part, this is because Marshall has stumbled into the same trap as Obama: talking too much about debt and deficits and too little about preventing House Budget Committee chair Paul Ryan, R-Wisconsin, from turning Medicare into a voucher program.

On the other hand, Republican Mark Amodei has barely noted Marshall’s presence in the race. He has run against Obama and House Democratic Leader Nancy Pelosi. “My opponent [is] trying very hard to nationalize the race,” says Marshall. “They’d like nothing better than to run against Nancy Pelosi and to run against the president, they don’t want to run against me.” True enough. And if the strategy succeeds, Republicans will keep right on running against Obama and Pelosi. Indeed, an Amodei win will steel GOP nerves for the fight against Obama’s jobs plan. There’s no question that a Marshall win would be an upset of significant proportions and signal that Democrats are in far better shape than even party’s strategists imagine. A Republican win would be less of a surprise. But it would signal that running against Obama and Pelosi works even in economically stressed areas.

BOTTOM LINE: Despite what the losers say, these special elections matter. If one party takes both races, there will be significant bragging rights. But even if the contests split along traditional lines—New York for Democrat Weprin, Nevada for Republican Amodei—the signals will be significant for short-term policy fights on the ground and for the 2012 electoral strategies of the Obama White House and the Congressional leaders of both parties.