Magnetic skyrmions (skyrmions hereafter) are magnetization configurations, whose topological robustness and nanoscale size have led to speculation that they could find use as a next-generation information carrier. Skyrmions have been observed in magnetic multilayer materials that are thin compared to the radius of a skyrmion, and chiral cubic single crystals that can be far larger than any characteristic skyrmion scale. In these single crystals, one would expect that skyrmions could exhibit interesting three-dimensional (3D) characteristics. Here, the symmetry of the micromagnetic free energy is investigated. This symmetry permits a complex 3D modulation of a skyrmion string, which we show to be a requirement of a skyrmion coexisting with the conical state. We discuss the implications of this modulation with respect to Thiele’s equation and interskyrmion interactions. Further to this internal modulation, we study theoretically and show experimentally that the strings themselves must contort towards the surfaces of their confining crystals.