Many good clubs and training organisations will provide a "right seat checkout". As others have said it is not a new rating and there is nothing to stop you flying right seat anyway.
However a checkout like this should include a variety of landings, some PFLs and general handling at the end of which you will be hopefully nearly as happy in the right as the left seat. If that is what you want it is not a bad idea to have this type of checkout if the object of the exercise is to be able to completely take over from the left seat passenger in any phase of the flight.
For many types the asi is in the "wrong" place from the right seat and aside form the other issues mentioned getting use to keeping a eye on the speed during the approach is vital. Flying on instruments from the right seat again because of the usual cockpit layout is also a new challenge.