P-values and e-values for non-parametric hypotheses with specified mean and variance

We build natural one-sided p-values and e-values for four types of non-parametric composite hypotheses with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality, and their combination. The baseline p-variable is roughly improved by a multiplicative factor of $1/2$ when assuming symmetry, $4/9$ when assuming unimodality, $2/9$ when assuming both. The baseline e-variable is improved by multiplicative factors of $2$, $1$ and $2$, respectively.