In this paper, a new pair of higher-order nondifferentiable multiobjective symmetric dual programs over arbitrary cones is formulated, where each of the objective functions contains a support function of a compact convex set. We identify a function lying exclusively in the class of higher-order K-η-convex and not in the class of K-η-bonvex function already existing in literature. Weak, strong and converse duality theorems are then established under higher-order K-η-convexity assumptions. Self duality is obtained by assuming the functions involved to be skew-symmetric. Several known results are also discussed as special cases