  We present an analysis of archival Chandra and XMM-Newton observations of the
magnetically-active cataclysmic variable DQ Her and the shell around it ejected
in a nova event in 1934. A careful revision of the Chandra observations
confirms previous claims on the presence of extended X-ray emission around DQ
Her and reveals that it actually corresponds to a bipolar jet-like structure
extending $\simeq$32$''$ along a direction from NE to SW. Therefore, this X-ray
emission extends beyond the optical nova shell and is perpendicular to its
major axis. The XMM-Newton observations confirm the presence of the extended
X-ray emission detected by Chandra, suggesting the additional presence of a
diffuse X-ray emission from a hot bubble filling the nova shell. This hot
bubble was very likely produced by the explosion that created the nebular shell
detected in optical images. The bipolar feature can be modelled by the
combination of an optically thin plasma emission component with temperature
$T\approx2\times10^{6}$ K and a power law component with a photon index of
$\Gamma=1.1\pm0.9$. Its X-ray luminosity in the 0.3 - 5 keV energy range is
$L_\mathrm{X}=(2.1\pm1.3)\times10^{29}$ erg s$^{-1}$, for an electron density
$n_\mathrm{e}\approx2$ cm$^{-3}$ and a mass $m_\mathrm{X}\approx
3\times10^{-6}$ M$_{\odot}$. We suggest that the X-ray bipolar structure in DQ
Her is a jet and interpret its non-thermal X-ray emission in terms of a
magnetized jet.
