Ordinary remodel and cleanout debris is fine. The limits come from California's hazardous waste rules and from Gardena's own code.
California treats batteries, fluorescent tubes and other mercury lamps, electronic devices, and thermostats as universal waste under the state Department of Toxic Substances Control, and none of those may go in ordinary trash. Paint, solvents, fuel, motor oil, other liquids, and asbestos stay out of a standard container as well.
Gardena takes this further than most cities. The franchised hauler is required to inspect what is put out and refuse anything it sees contaminated with hazardous substances, and to report reportable quantities to the state. That means a container can be turned down at the curb, not just at the gate, so sorting before you load genuinely matters here.
The city also keeps burying or depositing waste on any premises off limits, public or private, and its curbside bulky item service specifically does not cover construction and demolition debris, tires, or car bodies. Scrap metal, mirrors, ceramics, light bulbs, and construction materials are all excluded from the city recycling bins too.
Los Angeles County runs a free household hazardous and electronic waste program for items that cannot go in the trash, which is the route for leftover paint, chemicals, and old electronics. If an item has you guessing, ask your account manager before it goes in.