We study a sample of six X-ray-selected broad absorption line (BAL) quasi-stellar objects (QSOs) from the XMM-Newton Wide Angle Survey. All six objects are classified as BALQSOs using the classic balnicity index, and together they form the largest sample of X-ray-selected BALQSOs. We find evidence for absorption in the X-ray spectra of all six objects. An ionized absorption model applied to an X-ray spectral shape that would be typical for non-BAL QSOs (a power law with energy index α = 0.98) provides acceptable fits to the X-ray spectra of all six objects. The optical to X-ray spectral indices, αOX, of the X-ray-selected BALQSOs, have a mean value of <αOX> = 1.69 ± 0.05, which is similar to that found for X-ray-selected and optically selected non-BAL QSOs of a similar ultraviolet luminosity. In contrast, optically selected BALQSOs typically have much larger αOX and so are characterized as being X-ray weak. The results imply that X-ray selection yields intrinsically X-ray bright BALQSOs, but their X-ray spectra are absorbed by a similar degree to that seen in optically selected BALQSO samples; X-ray absorption appears to be ubiquitous in BALQSOs, but X-ray weakness is not. We argue that BALQSOs sit at one end of a spectrum of X-ray absorption properties in QSOs related to the degree of ultraviolet absorption in C IV 1550 Å.