Washington State’s primary elections on Tuesday offer a significant test of political polarization and President Trump’s influence.

In the Third Congressional District, Representative Marie Gluesenkamp Perez — a moderate Democrat who has regularly broken with her party to work across the aisle — is fighting off challengers from both the right and left to defend a rural swing district that voted three times for Mr. Trump. At the same time, in the conservative Fourth District in Central Washington, the retirement of Representative Dan Newhouse, a Republican, has touched off a scramble among G.O.P. contenders.

Under Washington’s nonpartisan primary system, all candidates appear on a single ballot regardless of party affiliation, with only the top two vote-getters advancing to the general election. Washington and California are the only states that use a nonpartisan “top-two” primary mechanism in congressional races.

Here’s what to watch:

A moderate Democrat faces rivals from the right and the left.

Ms. Gluesenkamp Perez’s seat is one of just 18 rated as tossups by the nonpartisan Cook Political Report, and is a top target for Republicans.

Mr. Trump has backed John Braun, the minority leader of the Washington State Senate and a retired Navy captain, betting that he is the best Republican to unseat Ms. Gluesenkamp Perez. Mr. Braun owns a business that manufactures emergency vehicles, and has campaigned heavily on economic issues.

Ms. Gluesenkamp Perez, a former co-owner of an auto shop who has sought to appeal to blue-collar voters, is also facing a serious challenge from Brent Hennrich, a progressive. But she is expected to advance to the general election.

In total, three Republicans, four Democrats, one independent and one member of the Cascade Party — a minor, centrist political party founded by the former Nirvana bassist Krist Novoselic — are running in the nonpartisan primary in the Third Congressional District, which touches the Pacific Ocean and the Oregon border.

The primary for a red seat will test Trump’s influence.

Mr. Newhouse’s retirement has opened his safely red seat in Central Washington for new blood after his more than 10 years in office. He was one of 10 House Republicans who voted to impeach the president after the Jan. 6, 2021, riot at the U.S. Capitol, putting him at odds with Mr. Trump.

Whoever replaces him is likely to be much more aligned with the president: Both Republicans running have courted Mr. Trump’s endorsement. The president is backing Amanda McKinney, a Yakima County commissioner, over Jerrod Sessler, an entrepreneur and former NASCAR driver who has run unsuccessfully in previous cycles.