  We study post-hoc ($e$-value-based) and post-hoc anytime valid inference for
testing exchangeability and general group invariance. Our methods satisfy a
generalized Type I error control that permits a data-dependent selection of
both the number of observations $n$ and the significance level $\alpha$. We
derive a simple analytical expression for all exact post-hoc valid $p$-values
for group invariance, which allows for a flexible plug-in of the test
statistic. For post-hoc anytime validity, we derive sequential $p$-processes by
multiplying post-hoc $p$-values. In sequential testing, it is key to specify
how the number of observations may depend on the data. We propose two
approaches, and show how they nest existing efforts. To construct good post-hoc
$p$-values, we develop the theory of likelihood ratios for group invariance,
and generalize existing optimality results. These likelihood ratios turn out to
exist in different flavors depending on which space we specify our alternative.
We illustrate our methods by testing against a Gaussian location shift, which
yields an improved optimality result for the $t$-test when testing sphericity,
connections to the softmax function when testing exchangeability, and an
improved method for testing sign-symmetry.
