  We study the phase diagram of magic-angle twisted symmetric trilayer graphene
in the presence of uniaxial heterostrain and interlayer displacement field. For
experimentally reasonable strain, our mean-field analysis finds robust Kekul\'e
spiral order whose doping-dependent ordering vector is incommensurate with the
moir\'e superlattice, consistent with recent scanning tunneling microscopy
experiments, and paralleling the behaviour of closely-related twisted bilayer
graphene (TBG) systems. Strikingly, we identify a new possibility absent in
TBG: the existence of $\textit{commensurate}$ Kekul\'e spiral order even at
zero strain for experimentally realistic values of the interlayer potential in
a trilayer. Our studies also reveal a complex pattern of charge transfer
between weakly- and strongly-dispersive bands in strained trilayer samples as
the density is tuned by electrostatic gating, that can be understood
intuitively in terms of the `cascades' in the compressibility of magic-angle
TBG.
