  Context. Scaling relations between cluster properties embody the formation
and evolution of cosmic structure. Intrinsic scatters and correlations between
X-ray properties are determined from merger history, baryonic processes, and
dynamical state.
  Aims. We look for an unbiased measurement of the scatter covariance matrix
between the three main X-ray observable quantities attainable in large X-ray
surveys -- temperature, luminosity, and gas mass. This also gives us the
cluster property with the lowest conditional intrinsic scatter at fixed mass.
  Methods. Intrinsic scatters and correlations can be measured under the
assumption that the observable properties of the intra-cluster medium hosted in
clusters are log-normally distributed around power-law scaling relations. The
proposed method is self-consistent, based on minimal assumptions, and requires
neither the external calibration by weak lensing, dynamical, or hydrostatic
masses nor the knowledge of the mass completeness.
  Results. We analyzed the 100 brightest clusters detected in the XXL Survey
and their X-ray properties measured within a fixed radius of 300 kpc. The gas
mass is the less scattered proxy (~8%). The temperature (~20%) is intrinsically
less scattered than the luminosity (~30%) but it is measured with a larger
observational uncertainty. We found some evidence that gas mass, temperature
and luminosity are positively correlated. Time-evolutions are in agreement with
the self-similar scenario, but the luminosity-temperature and the gas
mass-temperature relations are steeper.
  Conclusions. Positive correlations between X-ray properties can be determined
by the dynamical state and the merger history of the halos. The slopes of the
scaling relations are affected by radiative processes.
