The Centaur Industry
The wild stallion does not bother himself with the concept of a "sheath cleaning". He fucks. And that makes him cleaner than the saints of this earth. However, a girding, a castrated male, he does not fuck. Thus, he must be cleaned, lest his body be burned with the scourge of the earth, with rot and vitriol.
Now, here come the Centaur. The Centaur is half-human, half-horse. Now, how does a Centaur clean?
First, we must assume that the Centaur cannot clean himself with his hands. Second, we must assume that Centaur society values cleanliness more highly than in nature, and that Centaurs have stages to their hygiene, not dissimilar to a quick hand wash and a sterilization.
Sex
There is 1 major problem: STDs (search Erquine Venereal Disease). However, there are vaccines for Equine Herpesvirus 1 & 4 (the problem causing one, per wikipedia, is 3). There are none for Contagious Equine Metritis in real life. However, depending on the rough timeline of Centaur technologies, this can be a non-issue. Provided stringent vaccinations and international / distant travel regulations exists, the spread of EHV or CEM can be regulated to a functionally '0' level.
While a viable form of cleaning, as per wild horses, this has high potential to be 'not clean enough', as such, the second solution:
Showering
In modern society, cleaning is simple: Centaur Car Wash. It might also be practical to have specialized mounts designed for sheath cleaning. In personal homes, shower systems with variable pressure levels and adjustable heads (imagine a shower head you could turn a wheel to move up and down) could be used, with foam soap sprayers (or servants), but this would likely be reserved for rich fucks. More likely would be a communal Centaur Car Wash, where you walk at a set pace through a series of hoses at different pressures to clean your horse half, before showering your human half normally. You might even just scrub your head while going through.
The CCW would probably be taxpayer funded, tightly regulated, and include various levels of cleaning which a Centaur can pay for (a once daily or weekly basic pass would likely be issued and tied to one's ID).
I am too fucking tired to think about Centaur cock more.