  We present high-speed spectroscopic observations of the intermediate polar DQ
Herculis. Doppler tomography of two He I lines reveals a spiral density
structure in the accretion disc around the white dwarf primary. The spirals
look very similar to the spirals seen in dwarf novae during outburst. DQ Her is
the first well established intermediate polar in which spirals are seen, that
are in addition likely persistent because of the system's high mass transfer
rate. Spiral structures give an alternative explanation for sidebands of the WD
spin frequency that are found in IP light curves. The Doppler tomogram of He II
4686 indicates that a large part of the emission is not disc-like. Spin trails
of spectra reveal a pulsation in the He II 4686 emission that is believed to
result from reprocessing of X-rays from the white dwarf's magnetic poles in the
accretion flow close to the WD. We confirm the previous finding that the
pulsation is only visible in the red-shifted part of the line when the beam
points to the back side of the disc. The absence of reprocessed light from the
front side of the disc can be explained by obscuration by the front rim of the
disc, but the absence of extra emission from the blue-shifted back side of the
disc is puzzling. Reprocessing in accretion curtains can be an answer to the
problem and can also explain the highly non-Keplerian velocity components that
are found in the He II 4686 line. Our spin trails can form a strong test for
future accretion curtain models, with the possibility of distinguishing between
a spin period of 71s or 142s. Spin trails of data taken at selected orbital
phases show little evidence for a significant contribution of the bright spot
to the pulsations and allow us to exclude a recent suggestion that 71s is the
beat period and 70.8s the spin period.
