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Transport and diffusion of heat in one dimensional (1D) nonlinear systemswhich {\it conserve momentum} is typically thought to proceed anomalously.Notable exceptions, however, exist of which the rotator model is a prominentcase. Therefore, the quest arises to identify the origin of manifest anomaloustransport in those low dimensional systems. Here, we develop the theory forboth, momentum/heat diffusion and its corresponding momentum/heat transport. Wedemonstrate that the second temporal derivative of the mean squared deviationof the momentum spread is proportional to the equilibrium correlation of thetotal momentum flux. This result in turn relates, via the integrated momentumflux correlation, to an effective viscosity, or equivalently, to the underlyingmomentum diffusivity. We put forward the intriguing hypothesis that afluid-like momentum dynamics with a {\it finite viscosity} causes {\it normal}heat transport; its corollary being that superdiffusive momentum diffusion withan intrinsic {\it diverging viscosity} in turn yields {\it anomalous} heattransport. This very hypothesis is corroborated over wide extended time scalesby use of precise molecular dynamics simulations. The numerical validation ofthe hypothesis involves three distinct archetype classes of nonlinearpair-interaction potentials: (i) a globally bounded pair interaction (the notedrotator model), (ii) unbounded interactions acting at large distances (theFermi-Pasta-Ulam $\beta$ model, or the rotator model amended with harmonic pairinteractions) and (iii), a pair interaction potential being unbounded at shortdistances while displaying an asymptotic free part (Lennard-Jones model).

1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport