The frequently observed lopsidedness of the distribution of stars and gas in disc galaxies is still considered as a major problem in galaxy dynamics. It is even discussed as an imprint of the formation history of discs and the evolution of baryons in dark matter haloes. Here, we analyse a selected sample of 70 galaxies from the Westerbork HI Survey of Spiral and Irregular Galaxies. The HI data allow us to follow the morphology and the kinematics out to very large radii. In the present paper, we present the rotation curves and study the kinematic asymmetry. We extract the rotation curves of receding and approaching sides separately and show that the kinematic behaviour of disc galaxies can be classified by five different types: symmetric velocity fields where the rotation curves of receding and approaching sides are almost identical; global distortions where the rotation velocities of receding and approaching side have an offset which is constant with radius; local distortions which lead to large deviations in the inner and negligible deviations in the outer parts (and vice versa); and distortions which split the galaxies into two kinematic systems, visible in the different behaviour of the rotation curves of receding and approaching sides, which leads to a crossing and a change in side. The kinematic lopsidedness is measured from the maximum rotation velocities, averaged over the plateau of the rotation curves. This gives a good estimate of global lopsidedness in the outer parts of the sample galaxies. We find that the mean value of the perturbation parameter denoting the lopsided potential as obtained from the kinematic data is 0.056. 36% of all sample galaxies are globally lopsided, which can be interpreted as the disc responding to a halo that was distorted by a tidal encounter. In Paper II, we study the morphological lopsidedness for the same sample of galaxies.
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