HOUSTON — One candidate has called for the impeachment of President Obama. Another wants the National Guard to help secure the border. Yet another criticized the openly lesbian mayor here for marrying her longtime partner in Palm Springs, Calif., saying it was “part of a larger strategy of hers to turn Texas into California.” And all of the leading contenders want to allow Texans to carry handguns in holsters on their hips.

Four powerful state Republican officials have been locked in a tight race for lieutenant governor — a job that in Texas is no mere sinecure, but one with powers that rival the governor’s when it comes to controlling what comes out of the Legislature.

A month before the March 4 primary, the race is illustrating the increasing shift to the far right for Texas Republicans as the rivals — Lt. Gov. David Dewhurst, who is seeking re-election; State Senator Dan Patrick of Houston; Todd Staples, the agriculture commissioner; and Jerry Patterson, the land commissioner — try to appeal to the grass-roots and Tea Party conservatives who make up the bulk of the electorate in Republican primaries.

There has been much talk lately in national political circles about the demographic forces that might make Texas a competitive state for Democrats by 2020. But the battle to win the state’s No. 2 seat shows that much of Texas remains, for now, neither blue nor even purple, but a deep shade of red.