  We present the results from new deep {\it Chandra} ($\sim410$~ks) and {\it
Suzaku} ($\sim180$ ks) observations of the merging galaxy cluster
ZwCl\,0008.8+5215 ($z=0.104$). Previous radio observations revealed the
presence of a double radio relic located diametrically west and east of the
cluster center. Using our new {\it Chandra} data, we find evidence for the
presence of a shock at the location of the western relic, RW, with a Mach
number $\mathcal{M}_{S_X}=1.48^{+0.50}_{-0.32}$ from the density jump. We also
measure $\mathcal{M}_{T_X}=2.35^{+0.74}_{-0.55}$ and
$\mathcal{M}_{T_X}=2.02^{+0.74}_{-0.47}$ from the temperature jump, with {\it
Chandra} and {\it Suzaku} respectively. These values are consistent with the
Mach number estimate from a previous study of the radio spectral index, under
the assumption of diffusive shock acceleration ($\mathcal{M}_{\rm
RW}=2.4^{+0.4}_{-0.2}$). Interestingly, the western radio relic does not
entirely trace the X-ray shock. A possible explanation is that the relic traces
fossil plasma from nearby radio galaxies which is re-accelerated at the shock.
For the eastern relic we do not detect an X-ray surface brightness
discontinuity, despite the fact that radio observations suggest a shock with
$\mathcal{M}_{\rm RE}=2.2^{+0.2}_{-0.1}$. The low surface brightness and
reduced integration time for this region might have prevented the detection.
{\it Chandra} surface brightness profile suggests $\mathcal{M}\lesssim1.5$,
while {\it Suzaku} temperature measurements found
$\mathcal{M}_{T_X}=1.54^{+0.65}_{-0.47}$. Finally, we also detect a merger
induced cold front on the western side of the cluster, behind the shock that
traces the western relic.
