  Aims: We aim to study the temporal and spectral behaviour of the eclipsing
polar CSS081231:071126+440405 from the infrared to the X-ray regime.
  Methods: We obtained phase-resolved XMM-Newton X-ray observations on two
occasions in 2012 and 2013 in different states of accretion. In 2013 the
XMM-Newton X-ray and UV data were complemented by optical photometric and
spectroscopic observations.
  Results: CSS081231 displays two-pole accretion in the high state. The
magnetic fields of the two poles are 36 and 69 MG, indicating a non-dipolar
field geometry. The X-ray spectrum of the main accreting pole with the lower
field comprises a hot thermal component from the cooling accretion plasma,
$kT_{plas}$ of a few tens of keV, and a much less luminous blackbody-like
component from the accretion area with $kT_{\rm bb} \sim$ 50-100\,eV. The
high-field pole which was located opposite to the mass-donating star accretes
at a low rate and has a plasma temperature of about 4\,keV. At both occasions
the X-ray eclipse midpoint precedes the optical eclipse midpoint by 3.2
seconds. The center of the X-ray bright phase shows accretion-rate dependent
longitudinal motion of ~20 degrees.
  Conclusions: CSS081231 is a bright polar that escaped detection in the RASS
survey because it was in a low accretion state. Even in the high state it lacks
the prominent soft component previously thought ubiquitous in polars. Such an
excess may still be present in the unobserved extreme ultraviolet. All polars
discovered in the XMM-Newton era lack the prominent soft component. The
intrinsic spectral energy distribution of polars still awaits characterisation
by future X-ray surveys such as eROSITA. The trajectory taken by material to
reach the second pole is still uncertain.
