  We derive multiparty games that, if the winning chance exceeds a certain
limit, prove the incompatibility of the parties' causal relations with any
partial order. This, in turn, means that the parties exert a back-action on the
causal relations; the causal relations are dynamical. The games turn out to be
representable by directed graphs, for instance by an orientation of the
M\"obius ladder. We discuss these games as device-independent tests of
spacetime's dynamical nature in general relativity. To do so, we design a
relativistic setting where, in the Minkowski spacetime, the winning chance is
bound to the limits. In contrast, we find otherwise tame processes with
classical control of causal order that win the games deterministically. These
suggest a violation of the bounds in gravitational implementations. We obtain
these games by uncovering a "pairwise central symmetry" of the correlations in
question. This symmetry allows us to recycle the facets of the acyclic subgraph
polytope studied by Gr\"otschel, J\"unger, and Reinelt in the mid-80s for
combinatorial optimization. In addition, we derive multiparty games in a
scenario where the polytope dimension grows only linearly in the number of
parties. Here, exceeding the limits not only proves the dynamical nature of the
causal relations, but also that the correlations are incompatible with any
global causal order.
