  We examine the reconstruction of galaxy cluster radial density profiles
obtained from Chandra and XMM X-ray observations, using high quality data for a
sample of twelve objects covering a range of morphologies and redshifts. By
comparing the results obtained from the two observatories and by varying key
aspects of the analysis procedure, we examine the impact of instrumental
effects and of differences in the methodology used in the recovery of the
density profiles. We find that the final density profile shape is particularly
robust. We adapt the photon weighting vignetting correction method developed
for XMM for use with Chandra data, and confirm that the resulting Chandra
profiles are consistent with those corrected a posteriori for vignetting
effects. Profiles obtained from direct deprojection and those derived using
parametric models are consistent at the 1% level. At radii larger than
$\sim$6", the agreement between Chandra and XMM is better than 1%, confirming
an excellent understanding of the XMM PSF. We find no significant energy
dependence. The impact of the well-known offset between Chandra and XMM gas
temperature determinations on the density profiles is found to be negligible.
However, we find an overall normalisation offset in density profiles of the
order of $\sim$2.5%, which is linked to absolute flux cross-calibration issues.
As a final result, the weighted ratios of Chandra to XMM gas masses computed at
R2500 and R500 are r=1.03$\pm$0.01 and r=1.03$\pm$0.03, respectively. Our study
confirms that the radial density profiles are robustly recovered, and that any
differences between Chandra and XMM can be constrained to the $\sim$ 2.5%
level, regardless of the exact data analysis details. These encouraging results
open the way for the true combination of X-ray observations of galaxy clusters,
fully leveraging the high resolution of Chandra and the high throughput of XMM.
