  Photonic analogs of the moir\'e superlattices mediated by interlayer
electromagnetic coupling are expected to give rise to rich phenomena such as
nontrivial flatband topology. Here, we propose and demonstrate a scheme to tune
the flatbands in a bilayer moir\'e superlattice by employing the band offset.
The band offset is changed by fixing the bands of one slab while shifting those
of the other slab, which is accomplished by modifying the thickness of the
latter slab. Our results show that the band-offset tuning not only makes some
flatbands emerge and disappear, but also leads to two sets of flatbands that
are robustly formed even with the change of band offset over a broad range.
These robust flatbands form either at the AA-stack site or at the AB-stack
site, and as a result, a single-cell superlattice can support a pair of
high-quality localized modes with tunable frequencies. Moreover, we develop a
diagrammatic model to provide an intuitive insight into the formation of the
robust flatbands. Our work demonstrates a simple yet efficient way to design
and control complex moir\'e flatbands, providing new opportunities to utilize
photonic moir\'e superlattices for advanced light-matter interaction including
lasing and nonlinear harmonic generation.
