President Obama signs an executive order Tuesday designed to shed light on the pay practices of federal contractors. (Pete Marovich/Bloomberg)

Barack Obama was feelin’ good on Equal Pay Day, and I know this because he was droppin’ some g’s: “Republicans in Congress have been gummin’ up the works” on equal pay, the president said Tuesday, against a backdrop of nodding women.

“If you’ve got a daughter, if you’ve got a sister, if you’ve got a mom — and I know you’ve got a mom,’’ making sure women earn the same pay for the same work “is something you should care about,” he said.

No wonder he was jaunty, taunting his political adversaries to “prove me wrong” and “join us in the 21st century” by voting for the Paycheck Fairness Act in the Senate on Wednesday. Because that’s not going to happen.

But if such legislation is as redundant as its critics say it is, and women are already well insulated from wage discrimination, why aren’t Republicans eager to get in on his pitch to female voters?

Instead, conservatives continue to argue simultaneously that the wage gap is a myth and that the wage gap does exist — at the White House, that is.

“The White House is having a little problem on this themselves,” crowed House Majority Leader Eric Cantor (R-Va.), referring to reports indicating that the women of the West Wing make only 88 percent of what the president’s male employees earn.

So, should that problem be addressed, even if it is because more men are in more senior positions? Why wouldn’t Republicans vote yes and pass the popcorn?

The Republican National Committee referred to the legislation as a “desperate political ploy” that would make it impossible for employers to award merit pay, and would launch a slew of meritless lawsuits.

Desperate or not, it’s definitely good politics; the president and his party have shown that accusing Republicans of waging a “war on women” turns out the Democratic base; it worked for Obama in 2012, and even for Terry McAuliffe in last year’s gubernatorial election in Virginia.

The gap varies from study to study, of course; the White House says women on average make 77 percent of what a man in the same occupation does, while a recent New York City study showed that after age 35, women earn only about 78 percent of what men do in the same jobs.

Pooh-poohers of the wage gap try to explain it away by saying that women just work fewer hours, and according to an opinion piece in the Wall Street Journal, “gravitate toward jobs with fewer risks, more comfortable conditions, regular hours, more personal fulfillment and greater flexibility.” Which must explain why two-thirds of all minimum-wage workers are women and livin’ the dream.

But even the most conservative studies come up with about a 5 percent wage gap even after controlling for hours, education, experience and marital status. And are Republicans really arguing that 5 percent isn’t a sum worth quibbling over?

Lilly Ledbetter, the chronically underpaid former Goodyear worker whose name is on the 2009 law that gave employees more time to sue for pay discrimination, looked on as Obama signed two related executive orders Tuesday. One outlaws retaliation against employees of federal contractors who discuss their salaries — free speech for those who can’t necessarily afford campaign contributions, I guess — and the other directs the Labor Department to keep track of compensation paid by federal contractors by race and sex.

One Democratic female senator after another stepped to the microphone to make the case for paycheck fairness on the Senate floor Tuesday afternoon. They made no new arguments, but that didn’t matter.

Our mothers and grandmothers fought this same fight more than 50 years ago, said Sen. Kirsten Gillibrand (D-N.Y.), “but the promise made by the Fair Pay Act continues to be broken every day. . . . They’re stuck in the ‘Mad Men’ era,’’ she said of Republicans.

Sen. Amy Klobuchar (D-Minn.) at least tried to make her point in keeping with the season, saying that in her state, spring had finally sprung, meaning that “the snow will melt, the flowers will bloom, and it’s time to stop freezing the women out of the economy.”

Sen. Barbara Boxer (D-Calif.) displayed charts that showed all the things women could buy with the $11,000 more a man makes every year, on average, than a woman does — a year’s worth of groceries, rent or child care. Over a working life, that adds up to $443,000. With that kind of money, she said, a woman could pay off her mortgage or send three kids to the University of California — “a great school, I might say.”

When Sen. Barbara A. Mikulski (D-Md.), who led the effort, yielded time to a male colleague, Joe Manchin III (D-W.Va.), she joked that she should really only give him 77 percent of the two minutes the female senators got. Or about a minute and 45 seconds, she observed.

Seventy-seven percent is a “horrible, horrible statistic,’’ said Sen. Heidi Heitkamp, (D-N.D.), and that, at least, was one statement with which Republicans could agree.