  In the reconstruction of physics events at future e$^+$e$^-$ colliders the
calorimeter design has a crucial role in the overall detector performance. The
reconstruction of events with many jets in their final state sets stringent
requirements on the jet energy and angular resolutions. The energy resolution
for jets with energy of about 45 GeV is required to be at the 4-5\% level to
enable an efficient separation of the W and Z boson invariant masses. We
demonstrate in this paper how such a performance can be achieved by exploiting
a particle flow algorithm tailored for a hybrid dual-readout calorimeter made
of segmented crystals and fibers. The excellent energy resolution and linearity
of such calorimeter for both photons and neutral hadrons ($3\%/\sqrt{E}$ and
$26\%/\sqrt{E}$, respectively), inherent to the homogeneous crystals and
dual-readout technological choices, provides a powerful handle for the
development of a new approach for particle identification and jet
reconstruction. While the dual-readout particle flow algorithm (DR-PFA)
presented in this paper is at its early stage of development, it already
demonstrates the potential of a hybrid dual-readout calorimeter for jet
reconstruction by improving the jet energy resolution with respect to a
calorimeter-only reconstruction from 6.0\% to about 4.5\% for 45 GeV jets.
