  The tail process
$\boldsymbol{Y}=(Y_{\boldsymbol{i}})_{\boldsymbol{i}\in\mathbb{Z}^d}$ of a
stationary regularly varying random field
$\boldsymbol{X}=(X_{\boldsymbol{i}})_{\boldsymbol{i}\in\mathbb{Z}^d}$
represents the asymptotic local distribution of $\boldsymbol{X}$ 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 $Y_{\boldsymbol{i}}\to 0$ as
$|\boldsymbol{i}|\to\infty$ 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 $\boldsymbol{X}$. 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.
