Here’s how I think this thing is going to end.

On Tuesday, Hillary Rodham Clinton will win the Democratic Party’s presidential primary in West Virginia. But as with the victory that Gen. Andrew Jackson scored over the British in the 1815 Battle of New Orleans, it will be anticlimactic.

Jackson’s win came two weeks after the signing of a peace treaty ending the War of 1812. News of the treaty didn’t reach him until after the battle was over. Clinton’s win will come after just about every sober political analyst has conceded that Barack Obama will be the Democrat’s presidential nominee.

But that won’t stop the New York senator from continuing to campaign through May 20, when voters in Kentucky and Oregon cast their ballots in this drawn-out selection contest. She’ll stay in the race that long for one important reason: to be remembered for making her party count the votes cast in the Michigan and Florida primaries.

That’ll happen May 31 when the Democratic Party’s rules committee meets to decide whether to count the ballots cast by voters in those states — which violated party rules and held early primaries. Clinton beat Obama handily in Florida, where neither candidate campaigned, and won Michigan after Obama removed his name from that state’s ballot.

Clinton, who has long contended that voters shouldn’t be punished because state party officials violated the rules, put a punctuation mark on this argument in her Indiana victory speech.

“I am running to be president of all of America — North, South, East and West, and everywhere in between,” she said. “That’s why it is so important that we count the votes of Florida and Michigan. It would be a little strange to have a nominee chosen by (just) 48 states.”

Why does this matter to Clinton? Even if she were to get the lion’s share of the delegates from Michigan and Florida, she couldn’t overtake Obama’s lead among pledged delegates. So why obsess about getting these votes counted?

The answer, I think, is political legacy. Clinton wants to be remembered as the candidate who went down fighting for the rights of all Democrats to have their votes counted.

She wants to be known as someone who battled for the party faithful even when her own race was lost. She wants to be seen not as a candidate who stayed in the race too long and damaged Obama but as someone who stuck around until she got justice for voters who were being treated unjustly.

That’s not just altruism, it’s good politics.

It’s a good bet Democrats will increase their majorities this year in both houses of Congress. If Obama goes on to win the presidency, Clinton can be expected to carve out a leadership spot for herself as the Senate champion for women’s issues and the rights of working-class people. Making sure the Michigan and Florida votes are counted at the convention will enhance her claim to be a selfless champion for the rights of others.

If Obama loses in November to John McCain, the presumptive Republican presidential nominee, Clinton, not Obama, could emerge as the Democrats’ titular head — if she’s seen as a scrappy fighter who got the party to treat Florida and Michigan voters fairly.

This is why Clinton likely won’t quit the race before the end of the month. Over the next few weeks, she can rack up a couple more primary wins — and push the Democratic Party to give her a face-saving victory before the rules committee without doing great injury to Obama’s chances in November.

If this isn’t the reason Clinton remains in the race, it ought to be.