It is shown by example that a closed-loop system eigenvalue may be made totally insensitive by making the corresponding left eigenmode insensitive, even though no closed-loop system right eigenmode may be made insensitive. This paper, therefore, formulates and solves a fundamental response insensitivity problem. It is established to what extent output feedback control can be used to render the closed-loop system left eigenstructure insensitive to possibly many, not necessarily small, parameter variations in the open-loop state-space model. It is explained that the results presented cannot be seen merely by transposing all the results in a previous paper on the corresponding issues for the right eigenstructure. A lemma is given that details the loss of design freedom that occurs when a closed-loop eigenvalue is assigned to a transmission zero. Sufficient conditions for the total insensitivity of a closed-loop dyad mode are given. It is shown by example that more closed-loop eigenvalues may be made insensitive by a mixture of parametric left and right eigenstructure insensitivity than by either alone. Multistage parametric eigenstructure-insensitive design is discussed.