The tail process \(\varvec{Y}=(Y_{\varvec{i}})_{\varvec{i}\in \mathbb {Z}^d}\) of a stationary regularly varying random field \(\varvec{X}=(X_{\varvec{i}})_{\varvec{i}\in \mathbb {Z}^d}\) represents the asymptotic local distribution of \(\varvec{X}\) as seen from its typical exceedance over a threshold u as \(u\rightarrow \infty\). 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 spectral decomposition of the tail process, characterizes the class of all tail processes. We then restrict to the case when \(Y_{\varvec{i}}\rightarrow 0\) as \(|\varvec{i}|\rightarrow \infty\) 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 \(\varvec{X}\). 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 a typical cluster of extremes for moving average processes with random coefficients and heavy-tailed innovations.