While Jordan algebras are commutative, operator commutativity will behave badly for a general non-associative Jordan algebra. When the product operators of two elements commute, the elements are said to operator commute. In some Jordan algebras operator commutation can be badly behaved, for instance having elements a and b operator commute, while a2 and b do not operator commute. In this paper we study JB-algebras, real Jordan algebras which are also Banach spaces in a compatible manner. These include for instance the spaces of self-adjoint elements of unital C⁎-algebras. We show that elements a and b in a JB-algebra operator commute if and only if they generate an associative subalgebra of mutually operator commuting elements, and hence operator commutativity in JB-algebras is as well-behaved as it can be. Letting Ua denote the quadratic operator of a, we also show that positive a and b operator commute if and only if Uab2=Uba2. We use this result to conclude that the unit interval of a JB-algebra is a sequential effect algebra as defined by Gudder and Greechie.