Rhodium plated jewellery
Your ability to clean rhodium plated jewellery at home is very limited.
Rhodium is a noble metal from the platinum group of metals. It is a hard silver white metal and highly reflective. This makes it a perfect metal to plate jewellery with.
White gold and sterling silver can be plated with rhodium.
Why does white gold need to be rhodium plated?

White gold is plated with rhodium to give it a whiter silver colour finish and to conceal solder joins, which are often a different colour.
Gold is yellow, to make it white it is alloyed with either nickle or palladium. Some modern white gold alloys are very white. Most though are really either grey or a light brown.
Handmade white gold jewellery is often many pieces of metal soldered together. Often the settings are made of one alloy and other components, such as the shank of a ring, use a different alloy. These alloys will polish to different colours. Rhodium plating conceals this difference, giving the jewellery a consistent look.
Rhodium also gives the jewellery a harder scratch resistant finish.
Why is sterling silver rhodium plated?
Unlike white gold, sterling silver naturally polishes to a very white finish. Sterling silver, because it is alloyed with copper, oxidizes easily and quickly. Rhodium plating prevents this oxidization and also gives sterling silver a harder scratch resistant finish.

Not all sterling silver jewellery is rhodium plated. Rhodium plating is often reserved for designs which emulate more expensive white gold and diamond jewellery. Designs, like the one above, usually contain gem stones like cubic zirconias.
Rhodium plating is becoming a more common practice in large production sterling silver jewellery. It gives the large volume jewellery a very consistent finish and helps keep the jewellery looking fresh until it is purchased.
Cleaning rhodium plated jewellery at home.
At home you can improve the look of your rhodium plated jewellery by washing it in a jewellery cleaner like LilyJewels SparkleClean, either in an ultrasonic cleaner or by hand washing. Then give the jewellery a light polish with a soft polishing cloth like a Blue Sunshine polishing cloth.
Refinishing rhodium plated jewellery.
The jewellery must have any remaining original rhodium plating removed, normally by using emery paper. It is then re-polished. After thoroughly cleaning first with an ultrasonic cleaner, then electrolytic salts, it is often an acid dipped. The jewellery is given a new rhodium plating and then a final polish.

This is a process that simply can’t be done in your home. It requires skills and equipment not found outside jewellery workshops. Rhodium plating solutions are acid based and require an electroplating station like the one above. Precise temperatures, current and cleanliness are required for consistent results.
How often should you refinish rhodium plated jewellery?
Even though it is nice to have beautifully polished jewellery, presented in the condition it was when you first received it, I do not recommend re-rhodium plating jewellery to frequently. The refinishing process for rhodium plated jewellery is by necessity aggressive. Each time you have this jewellery refinished you are in fact having it reduced, as metal must be removed in the process.
Limit refinishing rhodium plated jewellery to once every two years.