  Recent studies establish that the cuprate pseudogap phase is susceptible at
low temperatures to forming not only a $d$-symmetry superconducting (SC) state,
but also a $d$-symmetry form factor (dFF) density wave (DW) state. The
concurrent emergence of such distinct and unusual states from the pseudogap
motivates theories that they are "intertwined" i.e derived from a quantum
composite of dissimilar broken-symmetry orders. Some composite order theories
predict that the balance between the different components can be altered, for
example at superconducting vortex cores. Here, we introduce sublattice
phase-resolved electronic structure imaging as a function of magnetic field and
find robust dFF DW states induced at each vortex. They are predominantly
unidirectional and co-oriented (nematic), exhibiting strong spatial-phase
coherence. At each vortex we also detect the field-induced conversion of the SC
to DW components and demonstrate that this occurs at precisely the eight
momentum-space locations predicted in many composite order theories. These data
provided direct microscopic evidence for the existence of composite order in
the cuprates, and new indications of how the DW state becomes long-range
ordered in high magnetic fields.
