An interpretation of operators' commutator is what happens when those operators' operations interfere with each other. This interference is what leads to the Uncertainty Principle. Like position and momentum operators. These operators measure those quantities, and attempting to do so for the same direction of position and momentum leads to interference. However, position and momentum in orthogonal directions do not interfere with each other.
another snazzy way to think about them is as indicators for "curvature". Since it suits a lot of people to think about physics as geometry, the commutation of operators can be used to see the curvature, more or less, of the "space" of states.
So basically we can infer either of the two quantities due to the commutative nature of the operators.Provided, they happen in the same direction , am I right to think it this way ?