  We report an observation of quantum oscillations (QOs) in the correlated
insulators with valley anisotropy of twisted double bilayer graphene (TDBG).
The anomalous QOs are best captured in the magneto resistivity oscillations of
the insulators at v = -2, with a period of 1/B and an oscillation amplitude as
high as 150 k{\Omega}. The QOs can survive up to ~10 K, and above 12 K, the
insulating behaviors are dominant. The QOs of the insulator are strongly D
dependent: the carrier density extracted from the 1/B periodicity decreases
almost linearly with D from -0.7 to -1.1 V/nm, suggesting a reduced Fermi
surface; the effective mass from Lifshitz-Kosevich analysis depends nonlinearly
on D, reaching a minimal value of 0.1 me at D = ~ -1.0 V/nm. Similar
observations of QOs are also found at v = 2, as well as in other devices
without graphite gate. We interpret the D sensitive QOs of the correlated
insulators in the picture of band inversion. By reconstructing an inverted band
model with the measured effective mass and Fermi surface, the density of state
at the gap, calculated from thermal broadened Landau levels, agrees
qualitatively with the observed QOs in the insulators. While more theoretical
understandings are needed in the future to fully account for the anomalous QOs
in this moir\'e system, our study suggests that TDBG is an excellent platform
to discover exotic phases where correlation and topology are at play.
