The flat bands that appear in some twisted van der Waals heterostructures provide a setting in which strong interactions between electrons lead to a variety of correlated phases 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 . In particular, heterostructures of twisted double bilayer graphene host correlated insulating states that can be tuned by both the twist angle and an external electric field 11 , 12 , 13 , 14 . Here, we report electrical transport measurements of twisted double bilayer graphene with which we examine the fundamental role of spontaneous symmetry breaking in its phase diagram. The metallic states near each of the correlated insulators exhibit abrupt drops in their resistivity as the temperature is lowered, along with associated nonlinear current–voltage characteristics. Despite qualitative similarities to superconductivity, the simultaneous reversals in the sign of the Hall coefficient point instead to spontaneous symmetry breaking as the origin of the abrupt resistivity drops, whereas Joule heating seems to underlie the nonlinear transport. Our results suggest that similar mechanisms are probably relevant across a broader class of semiconducting flat band van der Waals heterostructures.