  Temporal graphs are graphs where the edge set can change in each time step,
and the vertex set stays the same. Exploration of temporal graphs whose
snapshot in each time step is a connected graph, called connected temporal
graphs, has been widely studied. We extend the concept of graph automorphisms
from static graphs to temporal graphs and show that symmetries enable faster
exploration: We prove that a connected temporal graph with $n$ vertices and
orbit number $r$ (i.e., $r$ is the number of automorphism orbits) can be
explored in $O(r n^{1+\epsilon})$ time steps, for any fixed $\epsilon>0$. For
$r=O(n^c)$ for constant $c<1$, this is a significant improvement over the known
tight worst-case bound of $\Theta(n^2)$ time steps for arbitrary connected
temporal graphs. We also give two lower bounds for exploration, showing that
$\Omega(n \log n)$ time steps are required for some inputs with $r=O(1)$ and
that $\Omega(rn)$ time steps are required for some inputs for any $r$ with
$1\le r\le n$.
  The techniques we develop for fast exploration are used to derive the
following result for rendezvous in connected temporal graphs: Two agents are
placed by an adversary at arbitrary vertices and given full information about
the temporal graph, except that they do not have consistent vertex labels. The
agents can meet at a common vertex after $O(n^{1+\epsilon})$ time steps, for
any $\epsilon>0$. For some connected temporal graphs with constant orbit number
we present a complementary lower bound of $\Omega(n\log n)$ time steps.
Finally, we give a randomized algorithm to construct a temporal walk $W$ that
visits all vertices of a given orbit with probability at least $1-\epsilon$ for
any $0<\epsilon<1$ such that $W$ spans $O((n^{5/3}+rn)\log n)$ time steps. The
runtime of this algorithm consists of $O(n^{1/3} \log (n/\epsilon))$
linear-time scans of the snapshots that exist in this time span.
