  Composite topological phases with intriguing topology like M${\"o}$bius
strips emerge in sublattice symmetric non-Hermitian systems due to spontaneous
breaking of time-reversal symmetry at some parameter regime. While these phases
have been characterized by nonadiabatic complex geometric phases of multiple
participating complex bands, the physical properties of these phases largely
remain unknown. We explore the dynamical response of these phases by studying
Loschmidt echo from an initial state of the Hermitian Su-Schrieffer-Heeger
(SSH) model, which is evolved by a non-Hermitian SSH Hamiltonian after a sudden
quench in parameters. Topology-changing quenches display non-analytical
temporal behavior of return rates (logarithm of the Loschmidt echo) for the
non-Hermitian SSH Hamiltonian in the trivial, M${\"o}$bius and topological
phase. Moreover, the dynamical topological order parameter appears only at one
side of the Brillouin zone for the M${\"o}$bius phase case in contrast to both
sides of the Brillouin zone for quench by the trivial and topological phase of
the non-Hermitian SSH model. The last feature is a dynamical signature of
different symmetry constraints on the real and imaginary parts of the complex
bands in the M${\"o}$bius phase.
