WASHINGTON — The battle to shape the Republican Party’s direction after its electoral losses will see its first skirmish in the Capitol on Wednesday in a House leadership fight in which the profile might be low but the symbolism is high.

The House’s Republican leaders would dearly like to elevate Representative Cathy McMorris Rodgers of Washington to lead the House Republican Conference, putting a female face into the pantheon of the white male Republican leaders. But standing in their way is Representative Tom Price of Georgia, one of the most conservative members of the House, who has lined up some big guns in his quest for the fourth-ranking post in the House Republican conference.

The most important of those guns, Representative Paul D. Ryan of Wisconsin, the defeated Republican vice-presidential nominee, showed no sign of retreat Tuesday in a letter sent to colleagues endorsing Mr. Price.

“In the House of Representatives, we have an obligation and a real opportunity to produce and promote a positive and principled agenda for the American people,” Mr. Ryan wrote. “Over the past two years, we have offered bold solutions to our fiscal and economic challenges. We must not let up.”