For the WIS we have a nice decomposition into overprediction, underprediction and dispersion. We could do the same for CRPS, either via the WIS with all quantile levels between 0 and 1 or, for samples via the median forecast $m$ (thanks @jbracher):
dispersion as $D=\mathrm{CRPS}(F, m)$
overprediction as $O=0$ if $y\geq m$ and $O=\mathrm{CRPS}(F, y) - D$ if $y < m$
underprediction as $U=0$ if $y\leq m$ and $U=\mathrm{CRPS}(F, y) - D$ if $y > m$
For the WIS we have a nice decomposition into overprediction, underprediction and dispersion. We could do the same for CRPS, either via the WIS with all quantile levels between 0 and 1 or, for samples via the median forecast $m$ (thanks @jbracher):