  Two monolayers of graphene twisted by a small `magic' angle exhibit nearly
flat bands leading to correlated electronic states and superconductivity, whose
precise nature including possible broken symmetries, remain under debate. Here
we theoretically study a related but different system with reduced symmetry -
twisted {\em double} bilayer graphene (TDBLG), consisting of {\em two} Bernal
stacked bilayer graphene sheets, twisted with respect to one another. Unlike
the monolayer case, we show that isolated flat bands only appear on application
of a vertical displacement field $D$. We construct a phase diagram as a
function of twist angle and $D$, incorporating interactions via a Hartree-Fock
approximation. At half filling, ferromagnetic insulators are stabilized,
typically with valley Chern number $C_v=2$. Ferromagnetic fluctuations in the
metallic state are argued to lead to spin triplet superconductivity from
pairing between electrons in opposite valleys. Response of these states to a
magnetic field applied either perpendicular or parallel to the graphene sheets
is obtained, and found to compare favorably with a recent experiment. We
highlight a novel orbital effect arising from in-plane fields that can exceed
the Zeeman effect and plays an important role in interpreting experiments.
