We have to regard Heinlein's historical reference as to a time of this practice as 'roughly two centuries ago.' During the earliest period of the California Gold Rush, 1849 (to the early 1850s), it was not uncommon to ship dirty laundry to Hawaii. Obviously, one could not wait for the clothes to return (and sorting them out would be impossible. The way it worked was for a man to drop off a soiled shirt and to pick up a clean one at the same time. The ships traveling back and forth kept the cycle going. You'd never get your shirt back, but you were happy to have a clean shirt, and so that as sufficient.

This approach to the laundry industry testifies to how difficult it was to acquire the most basic of services in those first crazed years of the California Gold Rush. No one was available to launder the clothes that needed cleaning. The issue was a lack of workers since most wanted to go to the gold fields to pick up gold or they were employed in jobs more attractive than laundry work (which was considered the lowest of the low in the nineteenth-century West).

Edit: Thanks to the persistence of /u/scarlet_sage, I am amending what I have to saw here. Heinlein certainly heard about this practice, as the story seems to have had considerable circulation. It was either true, or it is one of the many aspects of Gold Rush-era folklore that has grown up around that remarkable period of world history.