I'd refine that and say, 1/3 of the voters are on the right and know it, 1/3 are on the left and know it, and 1/3 are rather lefty and don't know it.
When leftists appeal to the center and win they move the country sharply left (eg FDR). When the right appeals to the center and wins, the country moves slightly left. One could point to certain exceptions such as welfare reform, but to me this is just taking a loony left policy and making it somewhat more reasonable, i.e. less loony. The welfare and tax regimes we live under are essentially leftist.
The only way for republicans to force change is to occupy all 3 elective bodies and stack the judiciary with their people. Even so, I doubt that government will shrink. It's growth might slow perceptably, but that's about all we can hope for. The tendency to drift leftward won't be stopped until the electorate is re-educated--probably a multi-generational project.