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We present new X-ray observations of luminous heavily dust-reddened quasars (HRQs) selected from infrared (IR) sky surveys. HRQs appear to be a dominant population at high redshifts and the highest luminosities, and may be associated with a transitional 'blowout' phase of black hole and galaxy co-evolution models. Despite this, their high-energy properties have been poorly known. We use the overall sample of 10 objects with XMM–Newton coverage to study the high-energy properties of HRQs at $\langle$ Lbol $\rangle$  =1047.5 erg s−1 and $\langle$ z $\rangle$  =2.5. For seven sources with strong X-ray detections, we perform spectral analyses. These find a median X-ray luminosity of $\left\langle L_{\rm 2\!-\!10\, keV} \right\rangle = 10^{45.1}$  erg s−1, comparable to the most powerful X-ray quasars known. The gas column densities are NH = (1–8) × 1022 cm−2, in agreement with the amount of dust extinction observed. The dust-to-gas ratios are sub-Galactic, but are higher than found in local AGN. The intrinsic X-ray luminosities of HRQs are weak compared to the mid-IR ($L_{\rm 6\, \mu m}$) and bolometric luminosities (Lbol), in agreement with findings for other luminous quasar samples. For instance, the X-ray to bolometric corrections range from κbol ≈ 50 to 3000. The moderate absorption levels and accretion rates close to the Eddington limit ($\langle$ λEdd $\rangle$  =1.06) are in agreement with a quasar blowout phase. Indeed, we find that the HRQs lie in the forbidden region of the NH–λEdd plane, and therefore that radiation pressure feedback on the dusty interstellar medium may be driving a phase of blowout that has been ongoing for a few 105 yr. The wider properties, including [O iii] narrow-line region kinematics, broadly agree with this interpretation. [ABSTRACT FROM AUTHOR]