  The asymmetric distribution of chiral objects with opposite chirality is of
great fundamental interests ranging from molecular biology to particle physics.
In quantum materials, chiral states can build on inversion-symmetry-breaking
lattice structures or emerge from spontaneous magnetic ordering induced by
competing interactions. Although the handedness of a chiral state can be
changed through external fields, a spontaneous chirality flipping has yet to be
discovered. In this letter, we present experimental evidence of chirality
flipping via changing temperature in a topological magnet EuAl$_4$, which
features orthogonal spin and charge density waves (SDW/CDW). Using circular
dichroism of Bragg peaks in the resonant magnetic x-ray scattering, we find
that the chirality of the helical SDW flips through a first order phase
transition with modified SDW wavelength. Intriguingly, we observe that the CDW
couples strongly with the SDW and displays a rare
commensurate-to-incommensurate transition at the chirality flipping
temperature. Combining with first principles calculations and angle resolved
photoemission spectroscopy, we establish the Fermi surface origin of the
helical SDW with intertwined spin, charge, and lattice degrees of freedom in
EuAl$_4$. Our results reveal an unprecedented spontaneous chirality flipping
and lays the groundwork for a new functional manipulation of chirality through
momentum dependent spin-charge-lattice interactions.
