What is the correct way to clean spectrophotometry cuvettes?
I always managed to wash out the remaining solution with the corresponding solvent a couple of times, subsequently rinsing with acetone/isopropyl alcohol and leave it dry on air for a few minutes. The method of spectrophotometry requires that you are using homogeneous solution anyway, so there shouldn't be any precipitation/deposition, otherwise you cannot interpret the data with Beer-Lambert law. 

Also, don't soak cells in KOH-bath or similar alkaline media. And I'm always strongly against abrasives, especially when it comes to cleaning optics. Even quartz cuvettes can be occasionally damaged with the brush pick. Disposable PS-cells can be easily scratched. Plus, there is a little risk of hydraulic pressure knocking out the cuvette's bottom (like it happens with test tubes when cleaned too vigorously). Leave mechanical cleaning for the professors of mechanics *.

*From a story published in The Oxford Book of Scientific Anecdotes:


  Victor Moritz Goldschmidt, a distinguished geochemist, was reputed to have carried a capsule of potassium cyanide when he was planning his escape from Nazi Germany. When a friend in the engineering department of the university expressed interest, Goldschmidt was supposed to have replied that cyanide was for professors of chemistry; his friend, a professor of mechanics, would have to carry a rope.