Neutral diboson processes are precise probes of the Standard Model (SM) of particle physics, which entails high sensitivity to new physics effects. We identify in terms of dimension-8 effective operators the leading departures from the SM that survive in neutral diboson processes at high energy and that interfere with the unsuppressed SM helicity contributions. We describe symmetries and selection rules that single out those operators, both for weakly and strongly coupled physics beyond the SM. Finally, we show that unitarity and causality enforce, via dispersion relations, positivity constraints on the coefficients of these effective operators, reducing the parameter space which is theoretically allowed.