We investigate the behavior of probabilities of large deviations above the mean versus large deviations below the mean for random additive functionals in a variety of random media models. Among others, we treat the large deviations of the point-to-hyperplane first-passage percolation on Zd and both the last-passage and the first-passage oriented percolation in Zd-1 × Z+. We also consider the large deviations of the parabolic Anderson model with white noise potential with respect to the Lyapunov exponent of the model.