  A new scheme for producing semidiscrete self-trapped vortices
(\textquotedblleft swirling photon droplets\textquotedblright ) in photonic
crystals with competing quadratic ($\chi ^{(2)}$) and self-defocusing cubic
($\chi ^{(3)}$) nonlinearities is proposed. The photonic crystal is designed
with a striped structure, in the form of spatially periodic modulation of the
$\chi ^{(2)}$ susceptibility, which is imposed by the quasi-phase-matching
technique. Unlike previous realizations of semidiscrete optical modes in
composite media, built as combinations of continuous and arrayed discrete
waveguides, the semidiscrete vortex droplets are produced here in the fully
continuous medium. This work reveals that the system supports two types of
semidiscrete vortex droplets, \textit{viz}., onsite- and intersite-centered
ones, which feature, respectively, odd and even numbers of stripes,
$\mathcal{N}$. Stability areas for the states with different values of
$\mathcal{N}$ are identified in the system's parameter space. Some stability
areas overlap with each others, giving rise to multistability of states with
different $\mathcal{N}$. The coexisting states are mutually degenerate,
featuring equal values of the Hamiltonian and propagation constant. An
experimental scheme to realize the droplets is outlined, suggesting new
possibilities for the long-distance transmission of structured light carrying
orbital angular momentum in nonlinear media.
