We know that horses do not have control of their bowels when they march down the street. The question is... do centaurs?
Would it be a question of intelligence versus animal intelligence? Or is it a physiological issue?
closed as primarily opinion-based by Oblivious Sage, doppelgreener, GMJoe, SevenSidedDie, BESW Jun 5 at 4:24
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Actually, from a strictly real-world biological point of view, we know that horses can control their bowel movements - they have two anal sphincters, one under conscious control, just the same as humans. However, horses have two issues that lead to their defecating in the street:
These two points lead to horses defecating wherever and whenever bowel pressure dictates.
Since some people keep miniature horses in their houses and successfully train the aforementioned miniature horses to not defecate indoors (See here), it can be concluded that horses can be trained and maintain conscious control over their bowels. The average horse-in-the-street just doesn't care where it defecates - and has a physiological necessity to defecate frequently - every 45 minutes or so.
So, to answer your question, unless the centaurs in question want to act like animals, even in the absence of any centaur cultural requirement about location of defecation, there is no reason why a centaur in mixed-species company wouldn't be able to learn pretty quickly that defecating in public is a social no-no, and have the physiological equipment to prevent inadvertent defecation.
In addition, centaurs may have a physiology that includes higher-quality food than horses - a humanoid-sized dentition, even if modified for a herbivorous diet, would practically necessitate richer food - which would lead to a greater reduction in volume as the food travelled down the centaur's gut, thus necessitating less frequent defecation.