  The tail process $\bY=(Y_{\bi})_{\bi\in\Z^d}$ of a stationary regularly
varying time series or random field $\bX=(X_{\bi})_{\bi\in\Z^d}$ represents the
asymptotic local distribution of $\bX$ as seen from its typical exceedance over
a threshold $u$ as $u\to\infty$. Motivated by the standard Palm theory, we show
that every tail process satisfies an invariance property called
exceedance-stationarity and that this property, together with the polar
decomposition of the tail process, characterizes the class of all tail
processes. We then restrict to the case when $Y_{\bi}\to 0$ as $|\bi|\to\infty$
and establish a couple of Palm-like dualities between the tail process and the
so-called anchored tail process which, under suitable conditions, represents
the asymptotic distribution of a typical cluster of extremes of $\bX$. The main
message is that the distribution of the tail process is biased towards clusters
with more exceedances. Finally, we use these results to determine the
distribution of the typical cluster of extremes for moving average processes
with random coefficients and heavy-tailed innovations.
