  Moir\'e structures formed by twisting three layers of graphene with two
independent twist angles present an ideal platform for studying correlated
quantum phenomena, as an infinite set of angle pairs is predicted to exhibit
flat bands. Moreover, the two mutually incommensurate moir\'e patterns among
the twisted trilayer graphene (TTG) can form highly tunable moir\'e
quasicrystals. This enables us to extend correlated physics in periodic moir\'e
crystals to quasiperiodic systems. However, direct local characterization of
the structure of the moir\'e quasicrystals and of the resulting flat bands are
still lacking, which is crucial to fundamental understanding and control of the
correlated moir\'e physics. Here, we demonstrate the existence of flat bands in
a series of TTGs with various twist angle pairs and show that the TTGs with
different magic angle pairs are strikingly dissimilar in their atomic and
electronic structures. The lattice relaxation and the interference between
moir\'e patterns are highly dependent on the twist angles. Our direct spatial
mappings, supported by theoretical calculations, reveal that the localization
of the flat bands exhibits distinct symmetries in different regions of the
moir\'e quasicrystals.
