One of the central features of quantum theory is that there are pairs of quantum observables that cannot be measured simultaneously. This incompatibility of quantum observables is a necessary ingredient in several quantum phenomena, such as measurement uncertainty relations, violation of Bell inequalities, and steering. Two quantum observables that admit a simultaneous measurement are, in this respect, classical. A finer classification of classicality can be made by formulating four symmetric relations on the set of observables that are stronger than compatibility; they are broadcastability, one-side broadcastability, mutual nondisturbance, and nondisturbance. It is proven that the five relations form a hierarchy and their differences in terms of the required devices needed in a simultaneous measurement are explained. All four relations that are stronger than compatibility are completely characterized in the case of qubit observables.