New York’s highest court on Monday will weigh whether the Legislature has the power to overrule counties and dictate when and how local elections are handled.

At issue is the state’s Even Year Election Law, which took effect in January and moves many county and town elections to take place on even-year cycles alongside state and federal offices. Some local offices are exempt from the change, as are New York City and other large cities.

Eight Republican-leaning counties and individual voters want the court to declare the law violates the state constitution, which grants home rule powers to local governments—including …