The focus of this article is on the different behavior of large deviations of random subadditive functionals above the mean versus large deviations below the mean in two random media models. We consider the point-to-point first passage percolation time a(n) on Z(d) and a last passage percolation time Z(n). For these functionals, we have lim(n ->infinity) a(n)/n = v and lim(n ->infinity) Z(n)/n = mu. Typically, the large deviations for such functionals exhibits a strong asymmetry, large deviations above the limiting value are radically different from large deviations below this quantity. We develop robust techniques to quantify and explain the differences.