  At partial fillings of its flat electronic bands, magic-angle twisted bilayer
graphene (MATBG) hosts a rich variety of competing correlated phases that show
sample to sample variations. Divergent phase diagrams in MATBG are often
attributed to the sublattice polarization energy scale, tuned by the degree of
alignment of the hexagonal boron nitride (hBN) substrates typically used in van
der Waals devices. Unaligned MATBG exhibits unconventional superconductivity
and correlated insulating phases, while nearly perfectly aligned MATBG/hBN
exhibits zero-field Chern insulating phases and lacks superconductivity. Here
we use scanning tunneling microscopy and spectroscopy (STM/STS) to observe
gapped phases at partial fillings of the flat bands of MATBG in a new
intermediate regime of sublattice polarization, observed when MATBG is only
partially aligned ($\theta_{Gr-hBN}$ $\approx$ 1.65$^\circ$) to the underlying
hBN substrate. Under this condition, MATBG hosts not only phenomena that
naturally interpolate between the two sublattice potential limits, but also
unexpected gapped phases absent in either of these limits. At charge
neutrality, we observe an insulating phase with a small energy gap ($\Delta$ <
5 meV) likely related to weak sublattice symmetry breaking from the hBN
substrate. In addition, we observe new gapped phases near fractional fillings
$\nu$ = $\pm 1/3$ and $\nu$ = $\pm 1/6$, which have not been previously
observed in MATBG. Importantly, energy-resolved STS unambiguously identifies
these fractional filling states to be of single-particle origin, possibly a
result of the super-superlattice formed by two moir\'e superlattices. Our
observations emphasize the power of STS in distinguishing single-particle
gapped phases from many-body gapped phases in situations that could be easily
confused in electrical transport measurements.
