  The discovery of interaction-driven insulating and superconducting phases in
moir\'e van der Waals heterostructures has sparked considerable interest in
understanding the novel correlated physics of these systems. While a
significant number of studies have focused on twisted bilayer graphene,
correlated insulating states and a superconductivity-like transition up to 12 K
have been reported in recent transport measurements of twisted double bilayer
graphene. Here we present a scanning tunneling microscopy and spectroscopy
study of gate-tunable twisted double bilayer graphene devices. We observe
splitting of the van Hove singularity peak by ~20 meV at half-filling of the
conduction flat band, with a corresponding reduction of the local density of
states at the Fermi level. By mapping the tunneling differential conductance we
show that this correlated system exhibits energetically split states that are
spatially delocalized throughout the different regions in the moir\'e unit
cell, inconsistent with order originating solely from onsite Coulomb repulsion
within strongly-localized orbitals. We have performed self-consistent
Hartree-Fock calculations that suggest exchange-driven spontaneous symmetry
breaking in the degenerate conduction flat band is the origin of the observed
correlated state. Our results provide new insight into the nature of
electron-electron interactions in twisted double bilayer graphene and related
moir\'e systems.
