  Rhombohedral graphene multilayers provide a clean and highly reproducible
platform to explore the emergence of superconductivity and magnetism in a
strongly interacting electron system. Here, we use electronic compressibility
and local magnetometry to explore the phase diagram of this material class in
unprecedented detail. We focus on rhombohedral trilayer in the quarter metal
regime, where the electronic ground state is characterized by the occupation of
a single spin and valley isospin flavor. Our measurements reveal a subtle
competition between valley imbalanced (VI) orbital ferromagnets and intervalley
coherent (IVC) states in which electron wave functions in the two momentum
space valleys develop a macroscopically coherent relative phase. Contrasting
the in-plane spin susceptibility of the IVC and VI phases reveals the influence
of graphene's intrinsic spin-orbit coupling, which drives the emergence of a
distinct correlated phase with hybrid VI and IVC character. Spin-orbit also
suppresses the in-plane magnetic susceptibility of the VI phase, which allows
us to extract the spin-orbit coupling strength of $\lambda \approx 50\mu$eV for
our hexagonal boron nitride-encapsulated graphene system. We discuss the
implications of finite spin-orbit coupling on the spin-triplet superconductors
observed in both rhombohedral and twisted graphene multilayers.
