Wyoming, one of the most conservative states, has a legislature divided between a hard-right House and a merely right Senate. Ultraconservatives want a complete takeover in the Aug. 18 primaries.

© Will Warasila for The New York Times

Gustavo Gordillo, a leader of the New York D.S.A., wants to reach new voters. But in interviews with conservative media, the backlash can be sharp.

© Ahmed Gaber for The New York Times