Verizon wireless has rejected a request from NARAL to allow text messages to its customers.

I’m a Verizon customer and have had the text messaging feature turned off when someone was sending pornographic text messages to me, but I’m still glad they are not doing this at least for now.

Saying it had the right to block “controversial or unsavory” text messages, Verizon Wireless has rejected a request from Naral Pro-Choice America, the abortion rights group, to make Verizon’s mobile network available for a text-message program.

The other leading wireless carriers have accepted the program, which allows people to sign up for text messages from Naral by sending a message to a five-digit number known as a short code.

Text messaging is a growing political tool in the United States and a dominant one abroad, and such sign-up programs are used by many political candidates and advocacy groups to send updates to supporters.

But legal experts said private companies like Verizon probably have the legal right to decide which messages to carry. The laws that forbid common carriers from interfering with voice transmissions on ordinary phone lines do not apply to text messages.

The dispute over the Naral messages is a skirmish in the larger battle over the question of “net neutrality” — whether carriers or Internet service providers should have a voice in the content they provide to customers.

“This is right at the heart of the problem,” said Susan Crawford, a visiting professor at the University of Michigan law school, referring to the treatment of text messages. “The fact that wireless companies can choose to discriminate is very troubling.”

In turning down the program, Verizon, one of the nation’s two largest wireless carriers, told Naral that it does not accept programs from any group “that seeks to promote an agenda or distribute content that, in its discretion, may be seen as controversial or unsavory to any of our users.” Naral provided copies of its communications with Verizon to The New York Times.

In another paragraph the spokesman for Verizon says they may reconsider their position.