X-ray free-electron lasers are unique sources of high-brightness coherent radiation. However, existing devices supply only linearly polarized light, precluding studies of chiral dynamics. A device called the Delta undulator has been installed at the Linac Coherent Light Source (LCLS) to provide tunable polarization. With a reverse tapered planar undulator line to pre-microbunch the beam and the novel technique of beam diverting, hundreds of microjoules of circularly polarized X-ray pulses are produced at 500–1,200 eV. These X-ray pulses are tens of femtoseconds long, have a degree of circular polarization of 0.98–0.04+0.02 at 707 eV and may be scanned in energy. We also present a new two-colour X-ray pump–X-ray probe operating mode for the LCLS. Energy differences of ΔE/E = 2.4% are supported, and the second pulse can be adjusted to any elliptical polarization. In this mode, the pointing, timing, intensity and wavelength of the two pulses can be modified. Tunable polarization control and a two-colour X-ray pump–X-ray probe operating mode are demonstrated at the Linac Coherent Light Source (LCLS).