The different magnetic helicities conserved under conditions of perfect electrical conductivity are expressions of the fundamental property that every evolving fluid surface conserves its net magnetic flux. This basic hydromagnetic point unifies the well known Eulerian helicities with the Lagrangian helicity defined by the conserved fluxes frozen into a prescribed set of disjoint toroidal tubes of fluid flowing as a permanent partition of the entire fluid [B. C. Low, Astrophys. J. 649, 1064 (2006)]. This unifying theory is constructed from first principles, beginning with an analysis of the Eulerian and Lagrangian descriptions of fluids, separating the ideas of fluid and magnetic-flux tubes and removing the complication of the magnetic vector potential’s free gauge from the concept of helicity. The analysis prepares for the construction of a conserved Eulerian helicity, without that gauge complication, to describe a 3D anchored flux in an upright cylindrical domain, this helicity called absolute to distinguish it from the well known relative helicity. In a version of the Chandrasekhar-Kendall representation, the evolving field at any instant is a unique superposition of a writhed, untwisted axial flux with a circulating flux of field lines all closed and unlinked within the cylindrical domain. The absolute helicity is then a flux-weighted sum of the writhe of that axial flux and its mutual linkage with the circulating flux. The absolute helicity is also conserved if the frozen-in field and its domain are continuously deformed by changing the separation between the rigid cylinder-ends with no change of cylinder radius. This hitherto intractable cylindrical construction closes a crucial conceptual gap for the fundamentals to be complete at last. The concluding discussion shows the impact of this development on our understanding of helicity, covering (i) the helicities of wholly contained and anchored fields; (ii) the Eulerian and Lagrangian descriptions of field evolution; (iii) twist as a topological property of solenoidal fields versus the linkage properties of open and closed discrete curves treated by Gauss, Caligarneau, Berger, and Prior; and (iv) the change of absolute helicity by resistive diffusion. These are important hydromagnetic properties of twisted magnetic fields in the million-degree hot, highly conducting corona of the Sun.