  We investigate topological signatures in the short-time non-equilibrium
dynamics of symmetry protected topological (SPT) systems starting from initial
states which break the protecting symmetry. Na\"ively, one might expect that
topology loses meaning when a protecting symmetry is broken. Defying this
intuition, we illustrate, in an interacting Su-Schrieffer-Heeger (SSH) model,
how this combination of symmetry breaking and quench dynamics can give rise to
both single-particle and many-body signatures of topology. From the dynamics of
the symmetry broken state, we find that we are able to dynamically probe the
equilibrium topological phase diagram of a symmetry respecting projection of
the post-quench Hamiltonian. In the ensemble dynamics, we demonstrate how
spontaneous symmetry breaking (SSB) of the protecting symmetry can result in a
quantized many-body topological `invariant' which is not pinned under unitary
time evolution. We dub this `dynamical many-body topology' (DMBT). We show
numerically that both the pure state and ensemble signatures are remarkably
robust, and argue that these non-equilibrium signatures should be quite generic
in SPT systems, regardless of protecting symmetries or spatial dimension.
