New York (CNN Business) The ads submitted by women's sexual-health startup Dame Products to run in New York's transit system show female-focused sex toys.

But they don't look like anything specific — just smooth, if odd, shapes in shades of green, blue, maroon and light pink. One ad included the words, "Toys, for sex."

Dame says they were rejected by the Metropolitan Transportation Authority in December for promoting a "sexually oriented business."

Now, Dame is suing the MTA for rejecting its ads, alleging free speech violations by the agency, which is run by the state of New York. In its complaint, the company cites what it deems inconsistencies in ads that the MTA allows, including one featuring phallic-shaped cacti promoting a men's health startup that sells erectile dysfunction medication.

The startup says it spent roughly $150,000 to develop and revise the ad campaign it believed would be approved to run. But the MTA said in a memo to Dame that the proposed ads "promote a sexually oriented business, which has long been prohibited by the MTA's advertising standards." The memo is cited in a lawsuit filed on Tuesday by Dame in federal court in New York.

"The MTA's FAQs about its advertising policy clearly states that advertisements for sex toys or devices for any gender are not permitted, and advertising for FDA approved medication -- for either gender -- is permitted," MTA spokesman Maxwell Young said in a statement. "In its proprietary capacity as the operator of a transit system used by all New Yorkers, the MTA is constitutionally entitled to draw reasonable content-based distinctions between different types of advertisements and to consider its diverse customers."

One of the ads that sexual wellness startup Dame Products says was rejected by the MTA.

Dame says that prior to the rejection, it had collaborated with Outfront Media, the company that serves as the MTA's advertising contractor. After several months of back and forth, a representative for Outfront told Dame that it had "no objections" to two ads featuring its small, wearable vibrator called Fin, worn between the fingers, with two different text options: "Toys, for Sex" and "Get from Point A to Point O."

However, Outfront said some of the other ads submitted weren't suitable, according to the complaint. Outfront, which was not sued by Dame, declined to comment.

In late November, weeks after Dame submitted its revised ads based on the feedback, the same representative said the ads had been rejected and that the MTA would be releasing a new FAQ about its guidelines, according to the lawsuit. On December 3, the representative forwarded a memou from the MTA's chief development officer regarding its decision to not approve the ads. The updated FAQ specifies that "advertisements for sex toys or devices for any gender fall within" the prohibited "sexually oriented business" category.

"I think it's important that we are fair, and it is unfair that these other companies are being allowed to run advertisements that are validating sex in some way, and our company that targets female enjoyment is being viewed as inappropriate," Alexandra Fine, CEO and cofounder of Dame Products, told CNN Business in an exclusive interview.

The Brooklyn-based company, started by Fine and Janet Lieberman — an engineer who serves as cofounder and chief technology officer — aims to get people talking about consensual sex that is pleasurable for women.

There is a precedent for female-founded startups tackling culturally "taboo" topics pushing back against the MTA.

In 2015, Thinx , the period underwear company, was told by Outfront Media that its ads (featuring women in underwear, a half-grapefruit vaguely resembling a vagina, and dripping egg yolks) were inappropriate. Thinx went public with the rejection and leveraged the press to protest the decision. Its ads were ultimately approved.

Dame says it had reason to believe its ads would be approved, or that the MTA would find a way to work with it.

The MTA ran ads for birth control and libido medication for women, one of which read, "Men have 26 pills to make them hard. It's time we got ours."

Ads for another New York-based sex toy startup called Unbound were initially rejected last year, but the MTA did an about-face after the incident generated media attention. It told The New York Times it would find ways to work with Unbound on ads that don't violate its policies.

In Unbound's case, no ads ever ran: The startup was asked to remove "phallic symbols" from its ads in order to be approved, something that its CEO and cofounder Polly Rodriguez felt was a double standard, given it allows ads such as those selling erectile dysfunction products.

Rodriguez told CNN Business that what is being approved versus banned reaffirms the narrative that "male sexuality can be visible, and is deemed to health issue — and women's health is not to be seen in public, it's not to be visible. It's a really toxic narrative for us to have as a culture."

It's part of a broader uphill battle that startups tackling women's health and sexual wellness are facing when trying to enter into the mainstream.

In January, for instance, the Consumer Technology Association, which operates the Consumer Electronics Show, stripped Lora DiCarlo of an award for her hands-free vibrator innovation . The association said that "entries deemed by CTA in their sole discretion to be immoral, obscene, indecent, profane or not in keeping with CTA's image will be disqualified."

Four months later, after cries of gender bias, the industry group apologized and returned the award to DiCarlo.

The Dame complaint calls for an injunction, allowing Dame to display its ads on MTA property. It asks for the MTA to void its prohibition on ads promoting a "sexually oriented business" due to its "vagueness." It also asks for unspecified compensatory damages.

"In 2019, the MTA's Victorian view of female sexuality and the First Amendment cannot stand. ... All New Yorkers—and all women—deserve better," the complaint reads. "It reveals the MTA's sexism, its decision to privilege male interests in its advertising choices, and its fundamental misunderstanding of Dame's products."

The MTA's spokesman said in the statement that the company's "public statements are clearly inaccurate as the MTA's advertising is in no way gender-based or viewpoint discriminatory ... We intend to vigorously defend this lawsuit."