Mexi-Cali folk-rock group La Santa Cecilia has recorded with Rock Hall of Famer Elvis Costello, performed at the giant Bonnaroo festival and made a trip to the Grammy Awards podium. On Thursday, the quartet will make its Twin Cities debut at the posh new Ordway Concert Hall.

Only within the past year, however, have the group’s Mexican-American members been able to cross the border just 150 miles from their hometown of Los Angeles to perform in Mexico. That’s because one of the band members, accordionist Jose “Pepe” Carlos, emigrated illegally from his native Oaxaca, Mexico, at age 6 and was still fighting for citizenship 25 years later.

La Santa Cecilia raised the flag for immigration reform with its 2013 single “El Hielo (ICE),” a song detailing real-life experiences with U.S. Immigration and Customs Enforcement. The members spoke out for the cause again last year when it won the Grammy for best Latin rock album.

“We’re proud to be children of immigrant parents,” the group’s vivacious singer, Marisol Hernandez, yelled from the Grammy podium. “We dedicate this award to the more than 11 million undocumented people that live and work really hard in this country, and that still need to live a more dignified life in this country.”

Talking by phone last month from Los Angeles, La Santa Cecilia bassist Alex Bendana was happy to report that the group has since been able to perform in Mexico — as pleasant a surprise to the members as it is to come to northern states such as Minnesota.

“When we were in Milwaukee, we really didn’t know there was a big Latino population,” Bendana recalled. “We’re a bicultural band, so we we’re very proud to connect with Latinos all over the country. We all grew up in America and grew up on rock music, too, so, of course, we’re just as excited to play in other American cities for that reason.”

Offering an innovative blend of Mexican conjunto, bolero and norteño music, Southern soul and blues, and straight-up Americana rock, La Santa Cecilia has garnered high praise from English and Spanish language critics in recent years and become a popular attraction at cultural fests around North America.

The group broke through with its 2013 album, “Treinta Días,” thanks in part to Costello’s guest appearance in the song “Losing Game” (a producer friend turned him onto the band) and the subsequent Grammy. A lot of the attention also centered on the single “El Hielo,” whose video became an unofficial ad for the National Day Laborer Organizing Network.

“We were just telling our own stories and didn’t really know what effect the song would have,” Bendana said. “The characters in the song are real people. Eva in the song is Marisol’s mom, a domestic cleaner. Marta in the song is a friend of ours we did a benefit for to help her pay for her school.”

La Santa Cecilia got its start around Olvera Street, a historic section of downtown Los Angeles that dates to when the city was still in Mexico. As a young girl, Hernandez would serenade tourists with Mexican ballads outside her dad’s store. Most of the other band members cut their teeth playing mariachi tunes and at dances.

“For the band, it gave us an identity,” Bendana said. “Coming from Los Angeles has been a blessing to us. We have friends from all over the world here, including Latinos from throughout Latin America, plus Asians and African-Americans. It really feels like living in a melting pot.

“We can see it in other bands here, too. Bands like Los Lobos and Ozomatli sort of showed us how to represent the diverse side of this city.”

La Santa Cecilia put its culturally mish-mashed sound to the test last year when it finally made its Mexican debut at the Vive Latino Festival in Mexico City alongside the likes of such popular rock bands as Arcade Fire and Nine Inch Nails.

“We really didn’t know how people would react, because we were playing cumbias and norteño music, and they listen to that stuff all the time,” Bendana said. “Once we started playing, they understood what we are — we have that connection to Mexico, but we also have our American side, too. They seemed to appreciate that.”

Since then, the group has returned to Mexico for more shows. Bendana said those gigs have been particularly meaningful to the band’s accordionist, who finally earned U.S. residency status after he married last year.

“Imagine not being able to go to your native country for 26 years,” he said.

He recounted the struggles the group had in the United States trying to avoid immigration officials. “We’d have to take long detours to get to Texas. We couldn’t take I-10,” he said.

“It’s nice that’s all finally over for us, but of course it’d be nicer if all the other immigrants could be treated more fairly.”