Rumors were swirling across LGBT media on Monday about an imminent executive order targeting gays and lesbians.

“Anti-Gay Executive Order is Coming Soon,” warned gay blog JoeMyGod. “Possible Trump Action Specifically Targeting LGBTQ People,” screamed the media folks at the Human Rights Commission, America’s largest LGBT rights group.

Even the Washington Post weighed in with a tweet from reporter Josh Rogin “confirming” that a draft order “on LGBT issues including adoption” was well past the prep stage at the Trump White House.

It never materialized. In fact, by late Monday, the White House announced just the opposite — that it’s maintaining a landmark Obama-era executive order protecting LGBT federal workers and contractors. It was a rare moment of relief for progressive groups panicking over every potential new Trump proposal.

LGBT confusion over President Trump is understandable. On one hand, Trump has repeatedly declared he has little interest in rolling back LGBT rights — including marriage equality. Yet at the same time, key cabinet members, including Vice President Mike Pence, have long histories of enacting and supporting anti-LGBT legislation.

In truth, much like his predecessor, you might say Trump is “evolving.” And the lesson some LGBT groups seem unwilling or unable to learn is that they may actually be able to work with Trump. He has yet to suggest he wants to shut them out, so they shouldn’t shut him out, either.

True, Trump seems to favor religious-freedom acts exempting faith-based groups from serving gay customers or performing same-sex weddings. But he also opposed last year’s now-infamous anti-trans bathroom battles in North Carolina and was one of the first country club owners in snooty Palm Beach to admit gay couples.

It’s also worth pointing out that LGBT groups have a worrisome history of casting off new presidents far too early in their administrations.

Take Barack Obama — who left office as the most pro-LGBT president in US history. Barely six months into his presidency, a 2009 cover story in the LGBT magazine The Advocate brazenly declared that Obama had failed LGBTs — or at least hadn’t lived up to his promises. From activists to policy groups, progressives endlessly attacked Obama for failing to repeal the onerous Don’t Ask/Don’t Tell and the Defense of Marriage Act in his first year.

Both, of course, were actually enacted by Bill Clinton. And both, eventually, were dismantled under Obama — whose presidency also ushered in everything from marriage equality to hate-crime and equal-housing legislation.

Unlike Obama, Trump appears uninterested in traditional identity politics. While some may view this as a snub, such indifference actually positions ­LGBTs — at least for now — as an unlikely bridge between America’s outraged left and Trump’s right-wing policymakers.

Billionaire techie Peter Thiel, for instance, became the first openly gay man to ever address a Republican National Convention this summer, while Trump’s recent inauguration festivities featured a series of high-profile LGBT events. Gay groups like the Log Cabin Republicans and Gays for Trump were consistent fixtures at Trump rallies from primary season to election night.

Of course there’s been a lefty backlash. Critics have decried Thiel and his ilk as tokens at best — a fourth column at worst. Notable author and activist Kevin Sessums, for instance, has even dubbed them “Vichy Gays” — traitorous conspirators as indifferent to Trump-era injustices as the occupied French were to the perils of Nazism.

They’re “selfish in their bigotry,” Sessums says — both for championing a president with so many socially conservative associates and for failing to stand alongside other minority groups. These gays refuse to support “the others in society even when they are the others themselves,” Sessums explains.

But that’s shortsighted, and completely misunderstands the trend in gay rights. Throughout the Obama years, major LGBT groups focused on integrating gays openly in “conservative” institutions: marriage and the military.

So the presence of gay righties like Thiel in Trump’s circle may seem like an unorthodox alliance at first glance, but it isn’t. And instead of treating pro-Trump gays as collaborators with evil, they should be seen as what they are: representatives of the diverse interests of a diverse community.