Depends on the category. An applied mathematician is more "well rounded" IMHO than a theoretical physicist. An applied physicist is more "well rounded" than a theoretical mathematician. Here I define "well rounded" as "most able to easily get, and keep, a career outside of their field of expertise".
Having said that, from my own observation albeit limited, I'd speculate that the majority of math PhD's, maybe 70/30, are focused on more theoretical work versus applied, while the majority of physics PhD's (80/20) are focused on either solid-state, medical physics, experimental particle, etc. versus just theoretical work. This means from a numbers perspective, physicists in general would be more "well rounded", as experimental work usually demands one to be able to use many programming languages, as well as a bit more hands-on experience.
Keep in mind all numbers presented are completely made up and arbitrary, based on a loose "feeling" I have of a sample size of two universities. I myself am a theoretical physicst PhD, but I have a bachelors in applied physics, and minors in computational and statistal mathematics, and most of my electives were in electrical engineering. Thus, from a practical perspective I consider myself "well rounded", but I do not believe I would have a chance at an engineering position when put up against an equally skilled applied mathematician or experimental physicist.