Worker lawsuits are piling up against companies that let people use apps to summon shoppers, house cleaners, drivers and other services. The suits are all based on one crucial allegation that could rock the burgeoning on-demand market: Workers say they should be reclassified as employees, not independent contractors.

If the workers prevail in court, it would rewrite how a slew of companies do business, compelling them to pay overtime, minimum wage, workers’ compensation, unemployment insurance, break time and to reimburse for job-related expenses such as gasoline and auto insurance.

Cases filed this month go after housecleaning service Homejoy and delivery services Postmates and Try Caviar. Instacart’s grocery-delivery service was sued in January. Another case was filed late last year against home-repair and cleaning service Handy.com.

Instacart, Try Caviar (now owned by Square) and Homejoy declined to comment. Postmates did not respond to requests for comment.

Uber and Lyft drivers were the first on-demand workers to head to court to claim that they’re employees. Two San Francisco federal judges in the Northern District of California this month gave a boost to the drivers’ claims, which are proposed as class actions for all California drivers. The judges rejected Uber and Lyft’s attempts to have the lawsuits dismissed, and ruled that both suits have enough merit to be heard by juries.

“These companies thought that if they call themselves technology companies because they provide services by using a smartphone app, that somehow makes them different,” said Shannon Liss-Riordan, a Boston attorney who’s spearheading several of the cases. “They think they can get away with transferring the costs of doing business to their workers and depriving employees of the benefits they’re entitled to.”

She and other attorneys pursuing the suits say each company is in the business of providing a particular service — rides for Uber and Lyft, cleaning for Handy.com and Homejoy, deliveries for Postmates, Try Caviar and Instacart. That would mean the workers who carry out those services for customers are pivotal to the businesses and thus should be considered employees.

“Homejoy and Handy exert a lot of control over their workers,” said Byron Goldstein, an Oakland attorney suing those two companies, which offer both house-cleaning services and repair workers. “Homejoy workers cannot control the quality, timing or pay of their cleanings. They can’t increase or decrease their own profits.”

Claiming that technology makes a business exempt from 20th century laws “is a false narrative,” said Jonathan Davis, an attorney at San Francisco’s Arns Law Firm, which is suing Instacart. “Just because a worker is being controlled and directed by an algorithm and a smartphone is not different than being controlled and directed by a foreman.”

He sees it as a large economic issue, pointing to former Secretary of Labor Robert Reich’s term that the “share economy” should really be called the “share the scraps economy.”

“We’ve created a race-to-the-bottom economy where the responsibility for employment is shifted onto workers,” Davis said.

All of the startups being sued have deep pockets, with millions in venture backing. The attorneys suing them said they can afford to shell out more money to protect their workers.

“I would hope that these cases will bring a shift in the industry,” Liss-Riordan said. “I don’t see why companies can’t exist and succeed playing by the rules.”

Carolyn Said is a San Francisco Chronicle staff writer. E-mail: csaid@sfchronicle.com Twitter: @csaid