GLENDALE, Ariz. -- Arizona Gov. Jan Brewer says she meant no disrespect when she pointed a finger at President Barack Obama during an intense discussion on an airport tarmac. But the Republican governor said the Democratic president showed disrespect for her by abruptly ending their conversation.

The brief encounter -- out of earshot of observers but captured on camera -- was a highly visible demonstration of the verbal and legal skirmishing that has regularly occurred between Brewer and Obama's administration over illegal immigration and other issues.

Airport arrivals for presidents normally involve mere pleasantries between those involved, but Brewer and Obama have a history. And part of that history is what apparently got things going, according to accounts provided by Brewer and the White House.

Brewer said that during their talk, she invited Obama to visit Arizona to hear about her administration's achievements and to visit the U.S.-Mexico border, which has been a point of friction between the two because of the illegal immigration issue.

Obama then said Brewer's recently published book mischaracterized a 2011 White House meeting between them.

Brewer said in an interview Thursday, at another Phoenix-area airport, that she talks a lot with her hands and that her pointing a finger at Obama during their conversation wasn't disrespectful. "I respect the office of the president," she said. "I was there to welcome him."

She said she was grateful for the visit and intended to talk to him about the state's accomplishments. But she said she was "taken aback by his comments" when he said he wasn't happy with how her book described their White House meeting. Immediately after the meeting Brewer had said it was cordial, but her book said Obama lectured Brewer and that she felt he was condescending toward her.