Two distinct theoretical formalisms are developed for evaluating second derivatives of the energy analytically within the equation-of-motion coupled cluster approximation for excited electronic states (EOMEE-CC). In the first approach, both perturbations are treated equivalently. In the alternative formulation, the final operator expression is not symmetric with respect to interchange of the perturbations, and calculation of the second derivative requires that four systems of linear equations be solved for the first-order response of wave function parameters. However, only two systems need to be solved when the symmetric strategy is followed. While the symmetric approach superficially appears to be both more elegant and better suited for practical calculations, analysis shows that the former assertion is open to question and the latter only conditionally true. In particular, the asymmetric formulation is shown to be the preferred choice for all cases in which a large number of perturbations is involved. This is a rather important conclusion that holds not only for the EOMEE-CC method, but also for CC treatments of the electronic ground state and their finite-order many-body perturbation theory approximations.