Summary: Let \(f\) be an equilibrium bifunction defined on the product space \(\mathbb X\times \mathbb X\), where \(\mathbb X\) is a Banach space. If \(f\) is locally Lipschitz with respect to the second variable, for every \(x\in \mathbb X\) we define \(T_f(x)\) as the Clarke subdifferential of \(f(x,\cdot)\) evaluated at \(x\). This multivalued operator plays a fundamental role for the reformulation of equilibrium problems as variational inequality ones. We analyze additional conditions on \(f\) which ensure the \(D\)-maximal pseudomonotonicity and the cyclically pseudomonotonicity of \(T_f\). Such results have consequences in terms of the characterization of the set of solutions of a subclass of pseudomonotone equilibrium problems.