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We investigate the structure and velocity of cold molecular pillars, ``elephant trunks'', in expanding HII regions.The trunks are seen in silhouette against the bright background in our Hα images. All trunks are filamentary, and show signs of being twisted. Four such trunks in NGC 7822, IC 1805, the Rosette Nebula, and DWB 44 were selected, and then mapped mainly in ¹²CO and ¹³CO. We determine the mass and density of the trunks. Most of the mass is concentrated in a head facing the central cluster, and in sub-filaments forming the body of the trunk that is connected to V-shaped filaments to the outer expanding shell. We discovered that all four trunks rotate as rigid bodies (to a first approximation) about their major axes, and that at least two trunks are stretching along their major axes, meaning that the massive heads are lagging behind in the general expansion of the HII regions. The rotational periods are of the order of a few million years - similar to the age of the clusters. Rotation, then, is responsible for the twisted appearance of many elephant trunks, since they are rooted in the outer shells. The trunks carry surprisingly large amounts of angular momentum, 3x10⁴⁸-2x10⁵⁰kg.m²/s, with corresponding rotational energies of up to ∼10³⁷J. However, we estimate the total magnetic energies to be even larger. The trunks continuously reshape, and the formation of twined, and in many cases helical, sub-filaments can be understood as a consequence of electromagnetic and inertia forces inside the trunks. A theory based on the concept of magnetically twisted trunks is developed further, where the initial angular momentum is a consequence of the twisting of parent filaments containing mass condensations. Our results also suggest a new process of removing angular momentum from parent molecular clouds.